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User:R. J. Mathar/oeisPy/oeisPy/oeis core.py
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""" Native python3.1 implementations of OEIS http://oeis.org/classic functions. ==usage== python3 >>> from oeisPy import * The generic common format for function calls is Axxxxx(n) returns the n-th item of the sequence, where n is defined as usual depending on the offset. The first meaningful number is Axxxxxx(offset). Axxxxxx(n,k) returns the item in row n, column k of a triangle or array. Axxxxxx_list(n) returns a list with n items, starting with Axxxxxx(offset). Axxxxxx_row(n) returns all members of the n-th row of the sequence. The topmost row is obtained by Axxxxx_row(offset). isAxxxxxx(n) returns True or False depending on whether n is in the sequence or not. ==author== Richard J. Mathar, http://www.strw.leidenuniv.nl/~mathar/progs/oeisPy-0.0.1.tar.gz """ from nzmath import * import oeisPy.oeis_bulk as oeis_bulk import oeisPy.oeis_trans as oeis_trans def A000005(n): """ tau(.), the number of divisors of the argument. This function returns tau(n), the number of divisors of n. ==input== n -- positive integer ==output== integer -- function value ==examples== >>>oeis_core.A000005(6) 4 >>>oeis_core.A000005(100) 9 >>>oeis_core.A000005(51) 4 >>>oeis_core.A000005(1) 1 ==author== Richard J. Mathar 2010-06-29 """ return multiplicative.sigma(0,n) def A000005_list(n): """ The first n tau(.). This function returns the list of tau(1..n). ==input== n -- integer >=0. ==output== list -- The list starting [1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2..] ==examples== >>>oeis_core.A000005_list(6) [1, 2, 2, 3, 2, 4] ==author== Richard J. Mathar 2010-06-29 """ return [A000005(i) for i in range(1,n+1)] def A000009(n): """ Number of partitions into distinct parts. This function returns the number of partitions of the argument into distinct parts, or, the number of its partitions into odd parts. ==input== n -- integer >=0. ==output== integer -- function value ==examples== >>>oeis_core.A000009(0) 1 >>>oeis_core.A000009(6) 4 >>>oeis_core.A000009(26) 165 ==author== Richard J. Mathar 2010-07-03 """ # not implemented because some next() attribute isn't implemented in the nzmath library if False: f = {0: 1} # represents the polynomial 1 = 1*x**0 d = poly.formalsum.DictFormalSum(f,0) p = poly.uniutil.IntegerPolynomial(d,rational.theIntegerRing) for m in range(1,n+1): f = {0: 1, m : 1} d = poly.formalsum.DictFormalSum(f,0) pm = poly.uniutil.IntegerPolynomial(d,rational.theIntegerRing) p = p.ringmul(pm) return p[n] return A000009_list(n+1)[n] def A000009_list(n): """ List of number of partitions into distinct parts. ==input== n -- integer >=0. ==output== list -- The first n values of [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12..] ==examples== >>>oeis_core.A000009_list(0) [] >>>oeis_core.A000009_list(5) [1, 1, 1, 2, 2] ==author== Richard J. Mathar 2010-07-03 """ if n < 0: return [] a = [1,1,1] if n <=3 : return a[0:n] while len(a)<n: i = len(a) b = 0 for k in range(1,i+1): b += A000593(k)*a[i-k] a.append(b//i) return a def A000010(n): """ Euler's totient function phi(.). This function returns phi(n), the Euler totient function ==input== n -- positive integer ==output== integer -- function value ==examples== >>>oeis_core.A000010(11) 10 >>>oeis_core.A000010_list(10) [1, 1, 2, 2, 4, 2, 6, 4, 6, 4] ==author== Richard J. Mathar 2010-06-29 """ return multiplicative.euler(n) def A000010_list(n): return [A000010(i) for i in range(1,n+1)] def A000012(n): """ Constant 1. This function returns 1. ==input== n -- positive integer >=0 . ==output== integer -- function value ==examples== >>>oeis_core.A000012(11) 1 >>>oeis_core.A000012_list(10) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1] ==author== Richard J. Mathar 2010-06-29 """ return 1 def A000012_list(n): return [A000012(i) for i in range(n)] def A000027(n): """ Copy of the argument. This function returns the argument itself. ==input== n -- positive integer >=0 . ==output== integer -- n again. ==examples== >>>oeis_core.A000027(11) 11 >>>oeis_core.A000027_list(10) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] ==author== Richard J. Mathar 2010-06-29 """ return n def A000027_list(n): return [i for i in range(1,n+1)] def A000032(n): """ Lucas numbers. This function returns L(n), n-th Lucas number, starting 2, 1, 3, 4,... ==input== n -- integer >=0 . ==output== integer -- Lucas(n) ==examples== >>>oeis_core.A000032(0) 2 >>>oeis_core.A000032(1) 1 >>>oeis_core.A000032(2) 3 >>>oeis_core.A000032(3) 4 ==author== Richard J. Mathar 2010-07-01 """ return oeis_trans.linRec([2,1],[1,1],n) def A000032_list(n): """ List of Lucas numbers up to some small index. This function returns a list of the first Lucas(.) ==input== n -- integer >0 . ==output== list -- Lucas(0) up to Lucas(n-1) ==examples== >>>oeis_core.A000032_list(10) [2, 1, 3, 4, 7, 11, 18, 29, 47, 76] ==author== Richard J. Mathar 2010-07-01 """ return oeis_trans.linRec_list([2,1],[1,1],n) def isA000032(n): """ Test the argument for being a Lucas number. This function True or False depending on n being in A000032. ==input== n -- integer >=0 . ==output== boolean -- False for negative numbers, True for one of 1, 2, 3, 4, 7, 11 etc ==examples== >>>oeis_core.isA000032(0) False >>>oeis_core.isA000032(1) True >>>oeis_core.isA000032(2) True >>>oeis_core.isA000032(3) True ==author== Richard J. Mathar 2010-07-01 """ if n <= 0: return False if n < 5: return True L = [2,1,3] while L[2] < n: L[0:1] = [] L.append(L[1]+L[0]) return (L[2] ==n) def A000040(n): """ The prime at the index provided by the argument. This function returns the n-th prime ==input== n -- integer >=1 . ==output== integer -- prime(n) ==examples== >>>oeis_core.A000040(3) 5 >>>oeis_core.A000040_list(10) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] ==author== Richard J. Mathar 2010-06-29 """ return prime.prime(n) def A000040_list(n): """ The first n primes. This functin provides a list of the first n primes, starting [2,3,5,7,..] ==input== n -- integer >=1 . ==output== list -- The first n terms of [2,3,5,7,11,13,17,19,...] ==examples== >>>oeis_core.A000040_list(3) [2, 3, 5] """ # return [A000040(i) for i in range(1,n+1)] # the slow variant a =[] p = 2 while len(a) < n: a.append(p) p = prime.nextPrime(p) return a def isA000040(n): """ This function tests for primality. ==input== n -- integer >=1 . ==output== True or False ==examples== >>>oeis_core.isA000040(3) True >>>oeis_core.isA000040(890983) False ==author== Richard J. Mathar 2010-06-29 """ return prime.primeq(n) def A000041(n): """ This function returns the partition number of n ==input== n -- integer >=0 . ==output== integer -- number of partitions of n ==examples== >>>oeis_core.A000041(3) 3 ==author== Richard J. Mathar 2010-06-29 """ return combinatorial.partition_number(n) def A000041_list(n): """ List of parition numbers. This function returns the first n partition numbers. ==input== n -- integer >=0 . ==output== integer -- number of partitions of n ==examples== >>>oeis_core.A000041_list(10) [1, 1, 2, 3, 5, 7, 11, 15, 22, 30] ==author== Richard J. Mathar 2010-06-29 """ return [A000041(i) for i in range(n)] def A000043(n): """ Prime exponents of the n-th Mersenne prime. This function returns a prime p such that 2**p-1 is a Mersenne prime. This is not computed but simply taken from a static lookup table. ==input== n -- integer >0 . ==output== integer -- the prime exponent in the power of 2 that generates a Mersenne prime ==examples== >>>oeis_core.A000043(3) 5 >>>oeis_core.A000043(4) 7 ==author== Richard J. Mathar 2010-06-29 """ return A000043_list(n)[-1] def A000043_list(n): """ Prime exponents of the Mersenne primes. A list of prime epxonents that generate Mersenne primes. This is not computed but simply taken from a static lookup table. ==input== n -- integer >0 . ==output== list -- the first n prime exponents [2,3,5,7,13,..] ==examples== >>>oeis_core.A000043_list(10) [2, 3, 5, 7, 13, 17, 19, 31, 61, 89] ==author== Richard J. Mathar 2010-07-02 """ a = [2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917] if n <= len(a): return a[0:n] else: raise LookupError('Requested '+str(n)+' outsided tabulated range') def A000045(n): """ Fibonacci numbers. This function returns the n-th Fibonacci number ==input== n -- integer >=0 . ==output== integer -- Fibonacci(n) ==examples== >>>oeis_core.A000045(0) 0 >>>oeis_core.A000045(1) 1 >>>oeis_core.A000045(2) 1 ==author== Richard J. Mathar 2010-06-29 """ if n < 2: return n else: return A000045(n-1)+A000045(n-2) def A000045_list(n): """ List of the small Fibonacci numbers. This function returns the truncated list of the Fibonacci numbers 0,1,1,2,3,... ==input== n -- integer >=0 . ==output== list -- The first n values of [0,1,1,2,3,...] ==examples== >>>oeis_core.A000045_list(10) [0, 1, 1, 2, 3, 5, 8, 13, 21, 34] ==author== Richard J. Mathar 2010-06-29 """ return [A000045(i) for i in range(n)] def isA000045(n): """ This function tests for inclusion in the Fibonacci sequence ==input== n -- integer >=1 . ==output== True or False ==examples== >>>oeis_core.isA000045(3) True >>>oeis_core.isA000045(4) False >>>oeis_core.isA000045(890983) False >>>oeis_core.isA000045(39088169) True ==author== Richard J. Mathar 2010-06-29 """ if n < 0: return False elif n < 4: return True else: fo = 2 fim = 3 fi = 5 while fi <= n: if fi == n: return True fo = fim fim = fi fi += fo return False def A000079(n): """ This function returns the n-th power of 2. ==input== n -- integer >=0 . ==output== integer -- 2**n ==examples== >>>oeis_core.A000079(0) 1 >>>oeis_core.A000079(3) 8 >>>oeis_core.A000079_list(9) [1, 2, 4, 8, 16, 32, 64, 128, 256] ==author== Richard J. Mathar 2010-06-29 """ if n < 0: return 0 else: return 2**n def A000079_list(n): return [A000079(i) for i in range(n)] def isA000079(n): """ This function tests the argument against being a power of 2. ==input== n -- integer ==output== True or False ==examples== >>>oeis_core.isA000079(1) True >>>oeis_core.isA000079(4) True >>>oeis_core.isA000079(15) False ==author== Richard J. Mathar 2010-06-29 """ if n < 1: return False else: a = 0 nshft = n while nshft > 0: a += nshft % 2 nshft >>= 1 if a >1: return False return True def A000081(n): """ List of count of rooted trees with n nodes. ==input== n -- integer >=0 . ==output== integer -- the value ==examples== >>>oeis_core.A000081(6) 20 ==author== Richard J. Mathar 2010-07-03 """ return A000081_list(n+1)[-1] def A000081_list(n): """ List of count of rooted trees with k nodes, k =0 up to n-1 ==input== n -- integer >=0 . ==output== list -- The first n values of 0, 1, 1, 2, 4, 9, 20, 48, 115,... ==examples== >>>oeis_core.A000081_list(8) [0, 1, 1, 2, 4, 9, 20, 48] ==author== Richard J. Mathar 2010-07-03 """ a = [0,1,1] if n <= 3: return a[:n] while len(a)<n: anxt =0 i = len(a) for k in range(1,i): dsu = 0 for d in factor.misc.allDivisors(k): dsu += d*a[d] anxt += dsu*a[i-k] a.append(anxt// (i-1)) return a def A000108(n): """ Catalan(n). This function returns the n-th Catalan number ==input== n -- positive integer ==output== integer -- function value ==examples== >>>oeis_core.A000108(11) 58786 ==author== Richard J. Mathar 2010-06-29 """ return combinatorial.catalan(n) def A000108_list(n): """ List of first n Catalan numbers. This function returns the Catalan numbers starting with [1,1,2,...] ==input== n -- positive integer ==output== list -- the first n Catalan numbers ==examples== >>>oeis_core.A000108_list(4)) [1,1,2,5] ==author== Richard J. Mathar 2010-06-29 """ return [A000108(i) for i in range(n)] def A000110(n): """ Bell(n). This function returns the n-th Bell number. ==input== n -- positive integer ==output== integer -- function value ==examples== >>>oeis_core.A000110(0) 1 >>>oeis_core.A000110(1) 1 >>>oeis_core.A000110(11) 678570 ==author== Richard J. Mathar 2010-07-01 """ # if n ==0: # return 1 # a = 0 # for j in range(n): # a += A000166(j)* combinatorial.binomial(n-1,j) *(n-j)**(n-1) # return (a // combinatorial.factorial(n-1) ) return combinatorial.bell(n) def A000110_list(n): """ List of the first n Bell numbers. This function returns the Bell numbers starting with [1,1,2,5,15...] ==input== n -- positive integer ==output== list -- the first n Bell numbers ==examples== >>>oeis_core.A000110_list(6) [1,1,2,5,15,52] ==author== Richard J. Mathar 2010-07-01 """ return [A000110(i) for i in range(n)] def A000120(n): """ Hamming weight (number of 1's in the binary representation). This function returns Hamming(n). ==input== n -- integer >=0 . ==output== integer -- number of bits set in the binary representation. Set to zero for negative n. ==examples== >>>oeis_core.A000120(3) 2 >>>oeis_core.A000120(4) 1 ==author== Richard J. Mathar 2010-07-01 """ if n < 1: return 0 else: a = 0 nshft = n while nshft > 0: a += nshft & 1 nshft >>= 1 return a def A000120_list(n): """ List of the first binary weights. This function returns the binary weights starting with [0,1,1,2,1,...] ==input== n -- positive integer ==output== list -- the first n Hamming weights. ==examples== >>>oeis_core.A000120_list(7) [0, 1, 1, 2, 1, 2, 2] ==author== Richard J. Mathar 2010-07-01 """ return [A000120(i) for i in range(n)] def A000124(n): """ n*(n+1)/2+1. Lazy caterer's sequence at index n. ==input== n -- integer >=0 . ==output== integer -- 1+n*(n+1)/2 ==examples== >>>oeis_core.A000124(0) 1 >>>oeis_core.A000124(1) 2 >>>oeis_core.A000124(5) 16 ==author== Richard J. Mathar 2010-07-03 """ return 1+n*(n+1)//2 def A000124_list(n): """ The list of k*(k+1)/2+1 for k>=0. ==input== n -- integer >=0 . ==output== list -- The frist n term sof [1,2,4,7,11,..] ==examples== >>>oeis_core.A000124_list(5) [1, 2, 4, 7, 11] ==author== Richard J. Mathar 2010-07-03 """ return [1+i*(i+1)//2 for i in range(n)] def A000129(n): """ Pell numbers. This function returns Pell(n), n-th Pell number, starting 0, 1, 2, 5. ==input== n -- integer >=0 . ==output== integer -- Pell(n) ==examples== >>>oeis_core.A000129(0) 0 >>>oeis_core.A000129(1) 1 >>>oeis_core.A000129(2) 2 >>>oeis_core.A000129(6) 70 ==author== Richard J. Mathar 2010-07-01 """ return oeis_trans.linRec([0,1],[2,1],n) def A000129_list(n): """ List of Pell numbers up to some small index. This function returns a list of the first Pell(.) ==input== n -- integer >0 . ==output== list -- Pell(0) up to Pell(n-1) ==examples== >>>oeis_core.A000129_list(10) [0, 1, 2, 5, 12, 29, 70, 169, 408, 985] ==author== Richard J. Mathar 2010-07-01 """ return oeis_trans.linRec_list([0,1],[2,1],n) def isA000129(n): """ Test the argument for being a Pell number. This function True or False depending on n being in A000129. ==input== n -- integer >=0 . ==output== boolean -- False for negative numbers, True for one of 0,1,2,5,12,... ==examples== >>>oeis_core.isA000129(0) True >>>oeis_core.isA000129(1) True >>>oeis_core.isA000129(2) True >>>oeis_core.isA000129(3) False ==author== Richard J. Mathar 2010-07-01 """ if n < 0: return False if n < 3: return True P = [0,1,2] while P[2] < n: P[0:1] = [] P.append(2*P[1]+P[0]) return (P[2] ==n) def A000142(n): """ factorial This function returns n! ==input== n -- integer >=0 . ==output== integer -- factorial(n) ==examples== >>>oeis_core.A000142(3) 6 ==author== Richard J. Mathar 2010-06-29 """ return combinatorial.factorial(n) def A000166(n): """ Subfactorial(n) This function returns the n-th subfactorial. ==input== n -- nonnegative integer ==output== integer -- function value ==examples== >>>oeis_core.A000166(0) 1 >>>oeis_core.A000166(4) 9 ==author== Richard J. Mathar 2010-07-01 """ s = [1,0,1,2] if n <=3: return s[n] for i in range(4,n+1): s[0:1] = [] s.append(i*s[-1]+(-1)**i) return s[-1] def A000166_list(n): """ List of the first n subfactorials. This function returns the subfactorials starting with [1,0,1,2,9...] ==input== n -- positive integer ==output== list -- the first n Bell numbers ==examples== >>>oeis_core.A000166_list(6) [1, 0, 1, 2, 9, 44] ==author== Richard J. Mathar 2010-07-01 """ s = [1,0,1,2] if n <=4: return s[0:n] while len(s) < n: i = len(s) s.append(i*s[-1]+(-1)**i) return s def A000169(n): """ n**(n-1). This function returns the number of labelled rooted trees with n nodes. ==input== n -- integer >=1 ==output== integer -- function value ==examples== >>>oeis_core.A000166(1) 1 >>>oeis_core.A000166(5) 625 ==author== Richard J. Mathar 2010-07-17 """ return n**(n-1) def A000203(n): """ sigma(n) This function returns the sigma(n), the sum of divisors of n. ==input== n -- positive integer ==output== integer -- function value ==examples== >>>oeis_core.A000203(6) 12 ==author== Richard J. Mathar 2010-06-29 """ return multiplicative.sigma(1,n) def A000203_list(n): """ sigma(n) This function returns the a list of the first sigma(n). ==input== n -- positive integer ==output== list -- The truncated list of [1, 3, 4, 7,...] ==examples== >>>oeis_core.A000203_list(4) [1, 3, 4, 7] ==author== Richard J. Mathar 2010-06-29 """ return [A000203(i) for i in range(1,n+1)] def A000203(n): """ Lucas numbers. This function returns L(n), n-th Lucas number, starting 1, 3, 4,... ==input== n -- integer >0 . ==output== integer -- Lucas(n) ==examples== >>>oeis_core.A000203(1) 1 >>>oeis_core.A000203(2) 3 >>>oeis_core.A000203(3) 4 ==author== Richard J. Mathar 2010-07-01 """ if n <= 0: raise ValueError('non-positive argument '+str(n)) return A000032(n) def A000203_list(n): """ List of Lucas numbers up to some small index. This function returns a list of the first Lucas(.) ==input== n -- integer >0 . ==output== list -- Lucas(1) up to Lucas(n) ==examples== >>>oeis_core.A000203_list(9) [1, 3, 4, 7, 11, 18, 29, 47, 76] ==author== Richard J. Mathar 2010-07-03 """ a = A000032_list(n+1) return a[1:] def A000217(n): """ Triangular number. This function returns the n-th triangular number ==input== n -- integer >=0 . ==output== integer -- n*(n+1)/2 ==examples== >>>oeis_core.A000217(0) 0 >>>oeis_core.A000217(1) 1 >>>oeis_core.A000217(3) 6 ==author== Richard J. Mathar 2010-06-29 """ return n*(n+1)//2 def A000217_list(n): """ List of the first n Triangular numbers.. ==input== n -- integer >=0 . ==output== list -- n items starting [0, 1, 3, 10,..] ==examples== >>>oeis_core.A000217_list(10) [0, 1, 3, 6, 10, 15, 21, 28, 36, 45] ==author== Richard J. Mathar 2010-06-29 """ return [A000217(i) for i in range(n)] def isA000217(n): """ This function tests whether the argument is a triangular number ==input== n -- integer >=1 . ==output== True or False ==examples== >>>oeis_core.isA000217(3) True >>>oeis_core.isA000217(4) False >>>oeis_core.isA000217(120) True ==author== Richard J. Mathar 2010-06-29 """ if arith1.issquare(1+8*n) ==0: return False else: return True def A000225(n): """ 2**n-1. ==input== n -- integer >=0 . ==output== integer -- one less than the n-th power of 2. ==examples== >>>oeis_core.A000225(3) 7 >>>oeis_core.A000225(0) 0 ==author== Richard J. Mathar 2010-07-01 """ if n < 1: return 0 else: return 2**n -1 def A000244(n): """ The powers of 3. This function returns 3**n. ==input== n -- integer >=0. ==output== integer -- 3**n ==examples== >>>oeis_core.A000244(0) 1 >>>oeis_core.A000244(3) 27 >>>oeis_core.A000244(4) 81 ==author== Richard J. Mathar 2010-07-03 """ if n < 0: raise ValueError('negative argument '+str(n)) return 3**n def A000244_list(n): """ A list of the first n powers of 3. ==input== n -- integer >=0. ==output== list -- The list [1 ,3 ,9 ,27, 81,..] truncated to a length of n. ==examples== >>>oeis_core.A000244_list(8) [1, 3, 9, 27, 81, 243, 729, 2187] ==author== Richard J. Mathar 2010-07-03 """ return [3**i for i in range(n)] def isA000244(n): """ Test whether the argument is a positive power of 3. ==input== n -- integer >=0. ==output== boolean -- True if n is of the form 3**k, k>=0. ==examples== >>>oeis_core.isA000244(729) True >>>oeis_core.isA000244(8) False ==author== Richard J. Mathar 2010-07-03 """ if n <=0: return False e = 0 nshft = n while nshft >0: e += nshft % 3 if e > 1: return False nshft //= 3 return True def A000290(n): """ Squares This function returns the square of n. ==input== n -- integer ==output== integer -- n**2 ==examples== >>>oeis_core.A000290(0) 0 >>>oeis_core.A000290(-9) 81 ==author== Richard J. Mathar 2010-07-01 """ return n**2 def A000290_list(n): """ List of small squares This function returns the first n squares starting at 0. ==input== n -- integer ==output== list -- containing 0, 1, 4, 9 ,.. up to (n-1)**2 >>>oeis_core.A000290_list(10) [0, 1, 4, 9, 16, 25, 36, 49, 64, 81] ==author== Richard J. Mathar 2010-07-01 """ return [A000290(i) for i in range(n)] def isA000290(n): """ Test against being a perfect square. This function checks an integer argument for being an integer squared. ==input== n -- integer ==output== True or False ==examples== >>>oeis_core.isA000290(3) False >>>oeis_core.isA000290(-4) False >>>oeis_core.isA000290(9) True ==author== Richard J. Mathar 2010-07-01 """ if n ==0: return True if arith1.issquare(n) ==0: return False else: return True def A000292(n): """ Tetrahedral Numbers. This function returns n*(n+1)*(n+2)/6. ==input== n -- integer ==output== integer -- n*(n+1)*(n+2)/6. ==examples== >>>oeis_core.A000292(0) 0 >>>oeis_core.A000292(2) 4 >>>oeis_core.A000292(3) 10 ==author== Richard J. Mathar 2010-07-02 """ return n*(n+1)*(n+2)//6 def isA000292(n): """ Check the argument for being a tetrahedral number. This function is slow and parses the list of tetrahedral numbers with linear complexity in n. ==input== n -- integer ==output== boolean -- True if n can be written as k*(k+1)*(k+2)/6 with k>=0. ==examples== >>>oeis_core.isA000292(0) True >>>oeis_core.isA000292(2) False >>>oeis_core.isA000292(3) False ==author== Richard J. Mathar 2010-07-02 """ if n < 0: return False k = 0 t = 0 # t contains 6 times the k-th tetrahedral number while True: if t == 6*n: return True if t > 6*n: return False k += 1 t += 3*k*(k+1) def A000292_list(n): """ List of tetrahedral numbers. This function returns the first n tetrahedral numbers, starting at 0. ==input== n -- integer ==output== list -- containing n elements with 0, 1, 4, 10,.. >>>oeis_core.A000292_list(10) [0, 1, 4, 10, 20, 35, 56, 84, 120, 165] ==author== Richard J. Mathar 2010-07-02 """ return [A000292(i) for i in range(n)] def A000302(n): """ The powers of 4. This function returns 4**n. ==input== n -- integer >=0. ==output== integer -- 4**n ==examples== >>>oeis_core.A000302(0) 1 >>>oeis_core.A000302(3) 64 >>>oeis_core.A000302(4) 256 ==author== Richard J. Mathar 2010-07-03 """ if n < 0: raise ValueError('negative argument '+str(n)) return 4**n def A000302_list(n): """ A list of the first n powers of 4. ==input== n -- integer >=0. ==output== list -- The list [1, 4, 16, 64,... truncated to a length of n. ==examples== >>>oeis_core.A000302_list(8) [1, 4, 16, 64, 256, 1024, 4096, 16384] ==author== Richard J. Mathar 2010-07-03 """ return [4**i for i in range(n)] def isA000302(n): """ Test whether the argument is a positive power of 4. ==input== n -- integer >=0. ==output== boolean -- True if n is of the form 4**k, k>=0. ==examples== >>>oeis_core.isA000302(16484) True >>>oeis_core.isA000302(8) False ==author== Richard J. Mathar 2010-07-03 """ if n <=0: return False e = 0 nshft = n while nshft >0: e += nshft % 4 if e > 1: return False nshft //= 4 return True def A000326(n): """ Pentagonal numbers. This function returns n*(3n-1)/2. ==input== n -- integer ==output== integer -- n*(3*n-1)/2. ==examples== >>>oeis_core.A000292(0) 0 >>>oeis_core.A000292(1) 1 >>>oeis_core.A000292(5) 35 ==author== Richard J. Mathar 2010-07-03 """ return n*(3*n-1)//2 def isA000326(n): """ Test whether the argument is a pentagonal number. ==input== n -- integer ==output== boolean -- True if n is of the form k*(3k-1)/2, k>=0. ==examples== >>>oeis_core.isA000326(0) True >>>oeis_core.isA000326(2) False >>>oeis_core.isA000326(70) True ==author== Richard J. Mathar 2010-07-03 """ if n < 0: return False if n < 2: return True if arith1.issquare(1+24*n) ==0: return False else: return ((arith1.issquare(1+24*n)+1) %6) == 0 def A000330(n): """ Square pyramidal numbers. This function returns n*(n+1)*(2n+1)/6. ==input== n -- integer ==output== integer -- n*(n+1)*(2n+1)/6. ==examples== >>>oeis_core.A000330(0) 0 >>>oeis_core.A000330(2) 5 >>>oeis_core.A000330(3) 14 ==author== Richard J. Mathar 2010-07-03 """ return n*(n+1)*(2*n+1)//6 def isA000330(n): """ Check the argument for being a square pyramidal number. This function is slow and parses the list of numbers with linear complexity in n. ==input== n -- integer ==output== boolean -- True if n can be written as k*(k+1)*(2*k+1)/6 with k>=0. ==examples== >>>oeis_core.isA000330(0) True >>>oeis_core.isA000330(2) False >>>oeis_core.isA000330(55) True ==author== Richard J. Mathar 2010-07-02 """ if n < 0: return False k = 0 t = 0 # t contains 6 times the k-th square-pyramidal number while True: if t == 6*n: return True if t > 6*n: return False k += 1 t += 6*k**2 def A000330_list(n): """ List of tetrahedral numbers. This function returns the first n square-pyrimidal numbers, starting at 0. ==input== n -- integer ==output== list -- containing n elements with 0, 1, 5, 14,.. >>>oeis_core.A000330_list(10) [0, 1, 5, 14, 30, 55, 91, 140, 204, 285] ==author== Richard J. Mathar 2010-07-03 """ return [A000330(i) for i in range(n)] def A000043(n): """ Prime exponents of the n-th Mersenne prime. This function returns a prime p such that 2**p-1 is a Mersenne prime. This is not computed but simply taken from a static lookup table. ==input== n -- integer >0 . ==output== integer -- the prime exponent in the power of 2 that generates a Mersenne prime ==examples== >>>oeis_core.A000043(3) 5 >>>oeis_core.A000043(4) 7 ==author== Richard J. Mathar 2010-06-29 """ return A000043_list(n)[-1] def A000043_list(n): """ Prime exponents of the Mersenne primes. A list of prime epxonents that generate Mersenne primes. This is not computed but simply taken from a static lookup table. ==input== n -- integer >0 . ==output== list -- the first n prime exponents [2,3,5,7,13,..] ==examples== >>>oeis_core.A000043_list(10) [2, 3, 5, 7, 13, 17, 19, 31, 61, 89] ==author== Richard J. Mathar 2010-07-02 """ a = [2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917] if n <= len(a): return a[0:n] else: raise LookupError('Requested '+str(n)+' outsided tabulated range') def A000312(n): """ The power n**n. ==input== n -- integer >=0. ==output== integer -- n raised to its own power. ==examples== >>>oeis_core.A000312(2) 4 >>>oeis_core.A000312(4) 256 >>>oeis_core.A000312(0) 1 ==author== Richard J. Mathar 2010-07-03 """ if n <0: raise ValueError('non-positive argument '+str(n)) return n**n def A000364(n): """ The n-th secant number. Essentially the taylor coefficient [x^(2n)] sec(x) multiplied by (2n)!. ==input== n -- integer >=0. ==output== integer -- the n-th value in [1,1,5,61,1385,...] ==examples== >>>oeis_core.A000364(2) 3 >>>oeis_core.A000396(4) 1385 >>>oeis_core.A000396(14) 1252259641403629865468285 ==author== Richard J. Mathar 2010-07-17 """ return A000364_list(n+1)[-1] def A000364_list(n): """ The list of the first n secant numbers. The numerators of the coefficients of sec(x) = 1+x**2/2+5*x**4/24+.. ==input== n -- integer >=0. ==output== integer -- the first n values in [1,1,5,61,1385,...] ==examples== >>>oeis_core.A000364_list(4) [1, 1, 5, 61] ==author== Richard J. Mathar 2010-07-17 """ a = [1,1,5] while len(a) < n: anew = 0 nl = len(a) for i in range(nl): anew -= (-1)**(i+nl)*a[i]* combinatorial.binomial(2*nl,2*i) a.append(anew) return a[0:n] def A000396(n): """ The n-th perfect power. ==input== n -- integer between 0 and 14 (inclusive). ==output== integer -- n-th perfect power, starting 6, 28, 496,.. ==examples== >>>oeis_core.A000396(2) 28 >>>oeis_core.A000396(4) 8128 ==author== Richard J. Mathar 2010-07-03 """ return A000396_list(n)[-1] def A000396_list(n): """ A list of the smallest perfect powers. This is not computed but simply taken from a static lookup table. ==input== n -- integer >0 . ==output== list -- the first n perfect numbers in [2 6, 28 ,496,..] ==examples== >>>oeis_core.A000396_list(6) [6, 28, 496, 8128, 33550336, 8589869056] ==author== Richard J. Mathar 2010-07-03 """ a = [1, 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216, 13164036458569648337239753460458722910223472318386943117783728128, 14474011154664524427946373126085988481573677491474835889066354349131199152128, 23562723457267347065789548996709904988477547858392600710143027597506337283178622239730365539602600561360255566462503270175052892578043215543382498428777152427010394496918664028644534128033831439790236838624033171435922356643219703101720713163527487298747400647801939587165936401087419375649057918549492160555646976, 141053783706712069063207958086063189881486743514715667838838675999954867742652380114104193329037690251561950568709829327164087724366370087116731268159313652487450652439805877296207297446723295166658228846926807786652870188920867879451478364569313922060370695064736073572378695176473055266826253284886383715072974324463835300053138429460296575143368065570759537328128 ] if n <= len(a): return a[0:n] else: raise LookupError('Requested '+str(n)+' outsided tabulated range') def isA000578(n): """ Test the argument for being a positive perfect third power. ==input== n -- integer >=0 ==output== boolean -- True if the argument is >=0 and a perfect third power. ==examples== >>>oeis_core.isA000578(3) False >>>oeis_core.isA000578(8) True >>>oeis_core.isA000578(-8) False ==author== Richard J. Mathar 2010-07-03 """ if n < 0: return False if n < 2: return True f = factor.misc.FactoredInteger(n) for e in f.factors.values(): if ( e %3 ) != 0: return False return True def A000593(n): """ The sum of the odd divisors of the argument. ==input== n -- integer >=0. ==output== integer -- function value ==examples== >>>oeis_core.A000593(6) 4 >>>oeis_core.A000593(75) 124 ==author== Richard J. Mathar 2010-07-03 """ a = 0 for d in factor.misc.allDivisors(n): if (d &1) != 0: a += d return a def A000593_list(n): """ The list of the odd divisors of 1 to n. ==input== n -- integer >=0. ==output== list -- The truncated list starting [1, 1, 4, 1, 6, 4, 8, 1, 13, 6, 12, 4, ..] ==examples== >>>oeis_core.A000593_list(6) [1, 1, 4, 1, 6, 4] ==author== Richard J. Mathar 2010-07-03 """ return [A000593(i) for i in range(1,n+1)] def A000720(n): """ PrimePi, the number of primes less than or equal to the argument. This function returns the PrimePi(n). ==input== n -- integer ==output== integer -- pi(n) ==examples== >>>oeis_core.A000720(1) 0 >>>oeis_core.A000720(5) 3 ==author== Richard J. Mathar 2010-07-01 """ a = 0 p = 2 while p <= n: p = prime.nextPrime(p) a += 1 return a def A000720_list(n): """ List of PrimePi(.) up to small arguments. This function returns the first n values of PrimePi(.), starting with 0. ==input== n -- integer ==output== list -- containing 0, 1, 2, 2, 3, 3,... with n values. >>>oeis_core.A000720_list(10) [0, 1, 2, 2, 3, 3, 4, 4, 4, 4] ==author== Richard J. Mathar 2010-07-01 """ a = [] c = 0 for i in range(1,n+1): if prime.primeq(i): c += 1 a.append(c) return a def A000961(n): """ The n-th prime power. This function returns numbers of the form prime**e, e an integer >=0. For argument n=1, the 1 is returned. ==input== n -- integer >1 ==output== integer -- the n-th number of the prime**e format. ==examples== >>>oeis_core.A000961(1) 1 >>>oeis_core.A000961(2) 2 >>>oeis_core.A000961(18) 31 ==author== Richard J. Mathar 2010-07-02 """ if n <=0: raise ValueError('non-positive argument '+str(n)) i = 0 c = 0 while i < n: c += 1 while not isA000961(c): c += 1 i += 1 return c def A000961_list(n): """ The first n numbers which are prime powers. This function returns a list of numbers of the form prime**e, e an integer >=0, starting at 1. ==input== n -- integer >0 ==output== list -- The truncated list of [1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17...] ==examples== >>>oeis_core.A000961_list(1) [1] >>>oeis_core.A000961_list(2) [1,2] >>>oeis_core.A000961_list(9) [1, 2, 3, 4, 5, 7, 8, 9, 11] ==author== Richard J. Mathar 2010-07-02 """ a = [] c = 0 while len(a)<n: c += 1 while not isA000961(c): c += 1 a.append(c) return a def isA000961(n): """ Test n against being a prime power. This function returns True if n is of the form prime**e, e an integer >=0. ==input== n -- integer >0 ==output== boolen -- True if the argument is a prime power. ==examples== >>>oeis_core.isA000961(1) True >>>oeis_core.isA000961(2) True >>>oeis_core.isA000961(3) True >>>oeis_core.isA000961(4) True >>>oeis_core.isA000961(6) False ==author== Richard J. Mathar 2010-07-02 """ if n ==1: return True elif n <1: return False f = factor.misc.FactoredInteger(n) return (len(f.factors.values()) ==1 ) def A000984(n): """ The central binomial coefficient. This function returns the binomial(2*n,n) ==input== n -- integer >=0 ==output== integer -- C(2n,n) ==examples== >>>oeis_core.A000984(1) 2 >>>oeis_core.A000984(5) 252 ==author== Richard J. Mathar 2010-07-01 """ return combinatorial.binomial(2*n,n) def A001003(n): """ Little Schroeder numbers. This function returns the n-th Little Schroeder number ==input== n -- integer >=0 . ==output== integer -- The n-th element of [1,1,3,11,45,197,....] ==examples== >>>oeis_core.A001003(0) 1 >>>oeis_core.A001003(1) 1 >>>oeis_core.A001003(4) 11 >>>oeis_core.A001003(13) 71039373 ==author== Richard J. Mathar 2010-07-17 """ a = [1,1,3] if n <= 2: return a[n] for i in range(3,n+1): a[0:1] = [] a.append( ((6*i-3)*a[1]-(i-2)*a[0])//(i+1) ) return a[2] def A001003_list(n): """ The list of the first n Little Schroeder Numbers ==input== n -- integer >=0. ==output== list -- The truncated list starting [1, 1, 3, 11, 45, ...] ==examples== >>>oeis_core.A001003_list(6) [1, 1, 3, 11, 45, 197] ==author== Richard J. Mathar 2010-07-17 """ a = [1,1,3] while len(a)< n: i = len(a) a.append( ((6*i-3)*a[-1]-(i-2)*a[-2])//(i+1) ) return a[0:n] def A001006(n): """ Motzkin numbers. This function returns the n-th Motzkin number ==input== n -- integer >=0 . ==output== integer -- The n-th element of [1,1,2,4,9,21,....] ==examples== >>>oeis_core.A001006(0) 1 >>>oeis_core.A001006(1) 1 >>>oeis_core.A001006(4) 9 >>>oeis_core.A001006(29) 593742784858 ==author== Richard J. Mathar 2010-07-03 """ a = 0 for k in range(n+1): a += (-1)**(n-k)*combinatorial.binomial(n,k)*A000108(k+1) return a def A001006_list(n): """ List of the first Motzkin numbers. This function returns the list of the first n Motzkin numbers. ==input== n -- integer >=0 . ==output== integer -- The truncated variant of [1,1,2,4,9,21,....] ==examples== >>>oeis_core.A001006_list(5) [1, 1, 2, 4, 9] ==author== Richard J. Mathar 2010-07-03 """ a = [1,1,2] if n <= 3: return a[0:n] while len(a)< n: nexta = a[-1] i = len(a) for k in range(i-1): nexta += a[k]*a[i-2-k] a.append(nexta) return a def A001045(n): """ Jacobsthal numbers. This function returns J(n), n-th Jacobsthal number ==input== n -- integer >=0 . ==output== integer -- J(n) ==examples== >>>oeis_core.A001045(0) 0 >>>oeis_core.A001045(1) 1 >>>oeis_core.A001045(2) 1 >>>oeis_core.A001045(3) 3 ==author== Richard J. Mathar 2010-07-01 """ J = [0,1,1] if n <= 2: return J[n] for i in range(3,n+1): J[0:1] = [] J.append(J[1]+2*J[0]) return J[-1] def A001045_list(n): """ list of omega(n) This function returns a list of the first omega(.) ==input== n -- integer >0 . ==output== list -- omega(1) up to omega(n) ==examples== >>>oeis_core.A001045_list(10) [0, 1, 1, 3, 5, 11, 21, 43, 85, 171] ==author== Richard J. Mathar 2010-06-29 """ a = [0,1] if n <= 2: return a[0:n] while len(a) < n: a.append(a[-1]+2*a[-2]) return a def isA001045(n): """ Test the argument for being a Jacobstahl number. This function True or False depending on n being in A001045. ==input== n -- integer >=0 . ==output== boolean -- False for negative numbers, True for one of 0, 1, 3, 5, 11, 21 etc ==examples== >>>oeis_core.isA001045(0) True >>>oeis_core.isA001045(1) True >>>oeis_core.isA001045(2) False >>>oeis_core.isA001045(3) True ==author== Richard J. Mathar 2010-07-01 """ if n < 0: return False if n < 2: return True J = [0,1,1] while J[2] < n: J[0:1] = [] J.append(J[1]+2*J[0]) return (J[2] ==n) def A001060(n): """ Thue-Morse(n). ==input== n -- integer >=0 . ==output== integer -- 1 if the number of 1's in the binary representation of n is odd, 0 if it is even. ==examples== >>>oeis_core.A001045(0) 0 >>>oeis_core.A001045(1) 1 >>>oeis_core.A001045(2) 1 >>>oeis_core.A001045(3) 0 ==author== Richard J. Mathar 2010-07-01 """ return (A000120(n) & 1) def A001060_list(n): """ The first n values of the Thue-Mores sequence. ==input== n -- integer >=0 . ==output== list -- The first n values of [ 0, 1, 1, 0, 1, 0, 0, 1, 1, 0,..] ==examples== >>>oeis_core.A001060_list(10) [0, 1, 1, 0, 1, 0, 0, 1, 1, 0] ==author== Richard J. Mathar 2010-07-01 """ return [A001060(i) for i in range(n)] def A001147(n): """ Odd double factorial numbers. This function returns (2n-1)!!, the n-th odd double factorial. ==input== n -- integer >=0 . ==output== integer -- (2n-1)!!, defined as 1 for n=0. ==examples== >>>oeis_core.A001147(0) 1 >>>oeis_core.A001147(1) 1 >>>oeis_core.A001147(2) 3 >>>oeis_core.A001147(4) 105 ==author== Richard J. Mathar 2010-07-02 """ if n <0: raise ValueError('negative argument '+str(n)) a = 1 for m in range(3,2*n,2): a *= m return a def A001147_list(n): """ List of odd double factorial numbers. ==input== n -- integer >0 . ==output== list -- with n values starting [1,1,3,15,105,945,...] ==examples== >>>oeis_core.A001147_list(4) [1, 1, 3, 15] ==author== Richard J. Mathar 2010-07-02 """ a = [] if n >0: a = [1] m = 1 d = 1 while len(a) < n: a.append(d) m += 1 d *= 2*m-1 return a def A001157(n): """ sigma_2(n) This function returns the sigma_2(n), the sum of the squares of the divisors of n. ==input== n -- positive integer ==output== integer -- function value ==examples== >>>oeis_core.A001157(6) 50 ==author== Richard J. Mathar 2010-07-03 """ return multiplicative.sigma(2,n) def A001157_list(n): """ sigma(n) This function returns the a list of the first sigma_2(n). ==input== n -- positive integer ==output== list -- The truncated list of [[1, 5, 10, 21, 26, 50,..] ==examples== >>>oeis_core.A001157_list(4) [1, 5, 10, 21] ==author== Richard J. Mathar 2010-07-03 """ return [A001157(i) for i in range(1,n+1)] def A001221(n): """ omega(n) This function returns omega(n), the number of distinct prime factors of n ==input== n -- integer >0 . ==output== integer -- omega(n) ==examples== >>>oeis_core.A001221(0) 0 >>>oeis_core.A001221(1) 0 >>>oeis_core.A001221(30) 3 ==author== Richard J. Mathar 2010-06-29 """ f = factor.misc.FactoredInteger(n) return len(f.prime_divisors()) def A001221_list(n): """ list of omega(n) This function returns a list of the first omega(.) ==input== n -- integer >0 . ==output== list -- omega(1) up to omega(n) ==examples== >>>oeis_core.A001221_list(10) [0, 1, 1, 1, 1, 2, 1, 1, 1, 2] ==author== Richard J. Mathar 2010-06-29 """ return [A001221(i) for i in range(1,n+1)] def A001222(n): """ big-omega(n) This function returns big-omega(n), the number of prime factors of n counted with multiplicity ==input== n -- integer >0 . ==output== integer -- Omega(n) ==examples== >>>oeis_core.A001222(1) 0 >>>oeis_core.A001222(32) 5 ==author== Richard J. Mathar 2010-06-29 """ f = factor.misc.FactoredInteger(n) a = 0 for e in f.factors.values(): a += e return a def A001222_list(n): """ List of the first n values of big-omega(.) ==input== n -- integer >0 . ==output== list -- The list of [0, 1, 1, 2, 1, 2, 1, 3..] truncated to n terms. ==examples== >>>oeis_core.A001222_list(10) [0, 1, 1, 2, 1, 2, 1, 3, 2, 2] ==author== Richard J. Mathar 2010-06-29 """ return [A001222(i) for i in range(1,n+1)] def A001333(n): """ Numerators of the continued fraction convergents to sqrt(2). The n-th number in the list of a(0)=a(1) and a(n)=2*a(n-1)+a(n-2). ==input== n -- integer >=0 . ==output== integer -- The n-th term in [1, 1, 3, 7, 17, 41, 99, 239, 577, 1393..] ==examples== >>>oeis_core.A001333(0) 1 >>>oeis_core.A001333(1) 1 >>>oeis_core.A001333(2) 3 >>>oeis_core.A001333(6) 99 ==author== Richard J. Mathar 2010-07-03 """ return oeis_trans.linRec([1,1],[2,1],n) def A001333_list(n): """ List of the numerators of the continued fraction approximants to sqrt(2). ==input== n -- integer >0 . ==output== list -- The list of [1, 1, 3, 7, 17, 41, 99,..] truncated to n terms. ==examples== >>>oeis_core.A001333_list(10) [1, 1, 3, 7, 17, 41, 99, 239, 577, 1393] ==author== Richard J. Mathar 2010-07-03 """ return oeis_trans.linRec_list([1,1],[2,1],n) def isA001358(n): """ Test for being a semiprime Returns True if the argument is a semiprime, else False ==input== n -- integer >= 0 . ==output== boolean -- True if the argument is a semiprime, else false. ==examples== >>>oeis_core.isA001358(10) True >>>oeis_core.isA001358(11) False ==author== Richard J. Mathar 2010-07-01 """ if n < 4: return False else: return ( A001222(n) == 2 ) def A001358(n): """ semiprime(n) This function returns semprime(n), the n-th number which is a product of two primes ==input== n -- integer >0 . ==output== integer -- semiprime(n) ==examples== >>>oeis_core.A001358(1) 4 >>>oeis_core.A001358(3) 9 ==author== Richard J. Mathar 2010-07-01 """ c = 0 i = 3 while c < n: i += 1 while not isA001358(i): i += 1 c += 1 return i def A001358_list(n): """ list of low-indexed semiprimes This function returns a list of the first semiprime(.) ==input== n -- integer >0 . ==output== list -- semiprime(1) up to semiprime(n) ==examples== >>>oeis_core.A001358_list(10) [4, 6, 9, 10, 14, 15, 21, 22, 25, 26] ==author== Richard J. Mathar 2010-07-01 """ return [A001358(i) for i in range(1,n+1)] def A001405(n): """ The central binomial coefficient. This function returns binomial(n,floor(n/2)). ==input== n -- integer >=0 ==output== integer -- C(n,[n/2]) ==examples== >>>oeis_core.A001405(1) 1 >>>oeis_core.A001405(5) 10 ==author== Richard J. Mathar 2010-07-03 """ return combinatorial.binomial(n,n//2) def A001477(n): """ n . Note that there are easier ways to return the argument than calling this function :-) ==input== n -- integer ==output== integer -- n ==examples== >>>oeis_core.A005843(0) 0 >>>oeis_core.A005843(998) 998 ==author== Richard J. Mathar 2010-07-02 """ return n def A001478(n): """ Copy of the negative of the argument. This function returns the argument multiplied by -1. ==input== n -- positive integer >0 . ==output== integer -- -n ==examples== >>>oeis_core.A001478(11) -11 >>>oeis_core.A001478_list(10) [-1, -2, -3, -4, -5, -6, -7, -8, -9, -10] ==author== Richard J. Mathar 2010-09-06 """ return -n def A001478_list(n): return [-i for i in range(1,n+1)] def A001489(n): """ Copy of the negative of the argument. This function returns the argument multiplied by -1. ==input== n -- positive integer >=0 . ==output== integer -- -n ==examples== >>>oeis_core.A001489(11) -11 >>>oeis_core.A001489_list(10) [0, -1, -2, -3, -4, -5, -6, -7, -8, -9] ==author== Richard J. Mathar 2010-09-06 """ return -n def A001489_list(n): return [-i for i in range(0,n)] def A001519(n): """ The n-th term in a recurrence a(n)=3a(n-1)-a(n-2). Initialized with return values of 1 for 0<=n<=1. ==input== n -- integer >=0. ==output== integer -- a(n). ==examples== >>>oeis_core.A001519(0) 1 >>>oeis_core.A001519(1) 1 >>>oeis_core.A001519(4) 13 ==author== Richard J. Mathar 2010-07-03 """ return oeis_trans.linRec([1,1],[3,-1],n) def A001519_list(n): """ The first n terms of the recurrence a(n)=3a(n-1)-a(n-2). The first two elements are defined as 1. ==input== n -- integer >=0. ==output== list -- The first n elements of [1, 1, 2, 5, 13, 34, 89, 233,..] ==examples== >>>oeis_core.A001519(10) [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597] ==author== Richard J. Mathar 2010-07-03 """ return oeis_trans.linRec_list([1,1],[3,-1],n) def A001699(n): """ Number of binary trees of height n. ==input== n -- integer >= 0 ==output== integer -- The n-th element of [1, 1, 3, 21, 651, 457653,...] ==examples== >>>oeis_core.A001699(0) 1 >>>oeis_core.A001699(2) 3 >>>oeis_core.A001699(3) 21 ==author== Richard J. Mathar 2010-09-06 """ if n <0: raise ValueError('negative argument '+str(n)) return A001699_list(n+1)[-1] def A001699_list(n): """ Recursively a(n+1) = 2*a(n)*sum_{i=0..n-1}a(i) + a(n)^2. ==input== n -- integer >= 0 ==output== list -- The first n elements of [1, 1, 3, 21, 651, 457653,...] ==examples== >>>oeis_core.A001699_list(3) [1, 1, 3] ==author== Richard J. Mathar 2010-09-06 """ a = [1,1] while len(a) < n: i = len(a) s = 0 for j in range(i-1): s += a[j] s *= 2*a[-1] s += a[-1]**2 a.append(s) return a[0:n] def A001700(n): """ Ballot number binomial(2n+1,n+1). ==input== n -- integer ==output== integer -- C(2n+1,n+1). ==examples== >>>oeis_core.A001700(0) 1 >>>oeis_core.A001700(1) 3 >>>oeis_core.A001700(2) 10 ==author== Richard J. Mathar 2010-07-02 """ if n <0: return 0 else: return combinatorial.binomial(2*n+1,n+1) def A001700_list(n): """ Richard J. Mathar 2010-07-02 """ return [A001700(i) for i in range(n)] def A001764(n): """ binomial(3n,n)/(2n+1). Pfaff-Fuss-Catalan number C^3(n). ==input== n -- integer >=0 . ==output== integer -- C(3n,n)/(2n+1). ==examples== >>>oeis_core.A001764(0) 1 >>>oeis_core.A001764(1) 1 >>>oeis_core.A001764(13) 300830572 ==author== Richard J. Mathar 2010-07-03 """ if n <0: return 0 return combinatorial.binomial(3*n,n)//(2*n+1) def A001764_list(n): """ List of odd double factorial numbers. ==input== n -- integer >0 . ==output== list -- with n values starting [1,1,3,15,105,945,...] ==examples== >>>oeis_core.A001764_list(4) [1, 1, 3, 15] ==author== Richard J. Mathar 2010-07-02 """ a = [] if n >0: a = [1] m = 1 d = 1 while len(a) < n: a.append(d) m += 1 d *= 2*m-1 return a def A001906(n): """ Fibonacci number at index 2n. This function returns the Fibonacci(2n). ==input== n -- integer >=0 . ==output== integer -- Fibonacci(2*n) ==examples== >>>oeis_core.A001906(0) 0 >>>oeis_core.A001906(1) 1 >>>oeis_core.A001906(2) 3 >>>oeis_core.A001906(3) 8 >>>oeis_core.A001906(20) 102334155 ==author== Richard J. Mathar 2010-07-03 """ if n <0: raise ValueError('negative argument '+str(n)) return oeis_trans.linRec([0,1],[3,-1],n) def A001906_list(n): """ List of the small Fibonacci numbers. This function returns the truncated list of the bisected Fibonacci numbers 0 ,1 ,3, 8, 21,.. ==input== n -- integer >=0 . ==output== list -- The first n values of [0,1,1,2,3,...] ==examples== >>>oeis_core.A001906_list(10) [0, 1, 3, 8, 21, 55, 144, 377, 987, 2584] ==author== Richard J. Mathar 2010-07-03 """ return oeis_trans.linRec_list([0,1],[3,-1],n) def A002106(n): """ Number of transitive permutation groups of degree n. This is not computed but simply taken from a static lookup table. ==input== n -- integer >0 . ==output== integer -- The count of groups, including the primitive groups. ==examples== >>>oeis_core.A002106(3) 2 >>>oeis_core.A002106(6) 16 ==author== Richard J. Mathar 2010-09-06 """ return A002106_list(n)[-1] def A002106_list(n): """ Count of transitive permutation groups for degree >=1. This is not computed but simply taken from a static lookup table. ==input== n -- integer >0 . ==output== list -- the first n number of groups [1,1,2,5,5,..] ==examples== >>>oeis_core.A002106_list(10) [1, 1, 2, 5, 5, 16, 7, 50, 34, 45] ==author== Richard J. Mathar 2010-09-06 """ a = [1, 1, 2, 5, 5, 16, 7, 50, 34, 45, 8, 301, 9, 63, 104, 1954, 10, 983, 8, 1117, 164, 59, 7, 25000, 211, 96, 2392, 1854, 8, 5712, 12] if n <= len(a): return a[0:n] else: raise LookupError('Requested '+str(n)+' outsided tabulated range') def A002110(n): """ Primorial(n) This function returns the n-th primorial ==input== n -- integer >=0 . ==output== integer -- p# ==examples== >>>oeis_core.A002110(3) 30 ==author== Richard J. Mathar 2010-06-29 """ a = 1 for i in range(1,n+1): a *= prime.prime(i) return a def A002110_list(n): """ List of the first Primorials. This function returns the first n primorials ==input== n -- integer >=0 . ==output== list -- the first n primorials starting with 1. ==examples== >>>oeis_core.A002110_list(3) [1, 2, 6] ==author== Richard J. Mathar 2010-06-29 """ return [A002110(i) for i in range(n)] def isA002113(n): """ Test whether the argument is a palindrome in base 10. ==input== n -- integer >=0 . ==output== boolean -- True if the decimal representation is the same read forwards or backwards. ==examples== >>>oeis_core.isA002113(3) True >>>oeis_core.isA002113(303) True >>>oeis_core.isA002113(302) False ==author== Richard J. Mathar 2010-07-03 """ p = baseCoeff(n) l = len(p) for i in range( l//2): if p[i] != p[l-i-1]: return False return True def A002113_list(n): """ Return a list of the first palidromes in base 10. ==input== n -- integer >=0 . ==output== list -- the truncated list of [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66,..] ==examples== >>>oeis_core.A002113_list(10) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] ==author== Richard J. Mathar 2010-07-03 """ a = [] c = -1 while len(a)< n: c += 1 while not isA002113(c): c += 1 a.append(c) return a def A002113(n): """ Return the n-th palindrome in base 10. ==input== n -- integer >=0 . ==output== integer -- the n-th number (starting with 0) which is its own digit-reversed variant. ==examples== >>>oeis_core.A002113(1) 0 >>>oeis_core.A002113(20) 101 ==author== Richard J. Mathar 2010-07-03 """ i = 0 c = -1 while i< n: c += 1 while not isA002113(c): c += 1 i += 1 return c def A002275(n): """ Repunit(n) R_n, the number with decimal representation 1111....11 ==input== n -- integer >=0 . ==output== integer -- the number containing n ones. ==examples== >>>oeis_core.A002275(5) 11111 ==author== Richard J. Mathar 2010-06-29 """ if n <=0: return 0 else: return (10**n-1)//9 def A002275_list(n): """ List of repunit(n) ==input== n -- integer >=0 . ==output== list -- the first n repunits, 0, 1, 11, ..., R(n-1) ==examples== >>>oeis_core.A002275_list(4) [0, 1, 11, 111] ==author== Richard J. Mathar 2010-06-29 """ return [A002275(i) for i in range(n)] def isA002275(n): """ Test the argument for being a Repunit ==input== n -- integer >=0 . ==output== boolean -- True if the argument is 0 or of the form 1111...111 else False ==examples== >>>oeis_core.isA002275(111) True >>>oeis_core.isA002275(1110) False ==author== Richard J. Mathar 2010-07-01 """ if n <0: return False elif n ==0: return True else: nshft = n while nshft > 0: if ( nshft % 10 ) != 1: return False nshft //= 10 return True def A002378(n): """ Oblong number. This function returns n*(n+1). ==input== n -- integer ==output== integer -- n*(n+1) ==examples== >>>oeis_core.A002378(0) 0 >>>oeis_core.A002378(1) 2 >>>oeis_core.A002378(3) 12 ==author== Richard J. Mathar 2010-07-03 """ return n*(n+1) def A002378_list(n): """ List of the first n Oblong Numbers. ==input== n -- integer >=0 . ==output== list -- n items starting [0, 2, 6, 12,..] ==examples== >>>oeis_core.A002378_list(10) [0, 2, 6, 12, 20, 30, 42, 56, 72, 90] ==author== Richard J. Mathar 2010-07-03 """ return [i*(i+1) for i in range(n)] def isA002378(n): """ Test for being a number of the form k*(k+1). This function tests whether the argument is an Oblong Number. ==input== n -- integer >=1 . ==output== boolen -- False if the argument is not of the form k*(k+1), k>=0. ==examples== >>>oeis_core.isA002378(3) False >>>oeis_core.isA002378(6) True >>>oeis_core.isA002378(120) False ==author== Richard J. Mathar 2010-07-03 """ if n < 0: return False if arith1.issquare(1+4*n) ==0: return False else: return True def A002426(n): """ n-th Trinomial Coefficient. The largest coefficient in the expansion of (1+x+x^2)^n. ==input== n -- integer ==output== integer -- ==examples== >>>oeis_core.A002426(0) 1 >>>oeis_core.A002426(1) 1 >>>oeis_core.A002426(4) 19 ==author== Richard J. Mathar 2010-07-03 """ a=0 for k in range(n+1): # a += combinatorial.binomial(n,k)*combinatorial.binomial(k,n-k) if 2*k >= n: a += combinatorial.binomial(n,k)*combinatorial.binomial(k,n-k) return a def A002531(n): """ nth numerator of continued fraction convergents to sqrt(3). ==input== n -- integer >=0 . ==output== integer -- ==examples== >>>oeis_core.A002531(0) 1 >>>oeis_core.A002531(1) 1 >>>oeis_core.A002531(2) 2 >>>oeis_core.A002531(3) 5 >>>oeis_core.A002531(20) 262087 ==author== Richard J. Mathar 2010-09-06 """ if n <0: raise ValueError('negative argument '+str(n)) return oeis_trans.linRec([1,1,2,5],[0,4,0,-1],n) def A002531_list(n): """ n numerators of continued fraction convergents to sqrt(3). ==input== n -- integer >=0 . ==output== list -- starting [1,1,2,5,7,19..] ==examples== >>>oeis_core.A002531_list(3) [1, 1, 2] ==author== Richard J. Mathar 2010-09-06 """ if n <0: raise ValueError('negative argument '+str(n)) return oeis_trans.linRec_list([1,1,2,5],[0,4,0,-1],n) def A002620(n): """ floor(n^2/4). ==input== n -- integer ==output== integer -- n^2/4 rounded to zero. ==examples== >>>oeis_core.A002620(0) 0 >>>oeis_core.A002620(1) 0 >>>oeis_core.A002620(3) 2 ==author== Richard J. Mathar 2010-07-03 """ return (n**2)//4 def A002620_list(n): """ The values floor(k^2/4) starting at k=0. ==input== n -- integer ==output== list -- The first n elements of [0, 0, 1, 2, 4, 6, 9, 12, 16, 20..] ==examples== >>>oeis_core.A002620_list(10) [0, 0, 1, 2, 4, 6, 9, 12, 16, 20] ==author== Richard J. Mathar 2010-07-03 """ return [(i**2)//4 for i in range(n)] def A002658(n): """ Recursively a(n+1) = a(n)*sum_{i=0..n-1}a(i)+ a(n)*(a(n)+1)/2. ==input== n -- integer >= 0 ==output== list -- The a(n) elements of a(0)=a(1)=1, a(2)=2, a(3)=7... ==examples== >>>oeis_core.A002658(6) 2595782 ==author== Richard J. Mathar 2010-09-06 """ return A002658_list(n+1)[-1] def A002658_list(n): """ Recursively a(n+1) = a(n)*sum_{i=0..n-1}a(i)+ a(n)*(a(n)+1)/2. ==input== n -- integer >= 0 ==output== list -- The first n elements of [1, 1, 2, 7, 56,..] ==examples== >>>oeis_core.A002658_list(7) [1, 1, 2, 7, 56, 2212, 2595782] ==author== Richard J. Mathar 2010-09-06 """ a = [1,1,2] while len(a) < n: i = len(a) s = 0 for j in range(i-1): s += a[j] s *= a[-1] s += a[-1]*(1+a[-1])//2 a.append(s) return a[0:n] def A002808(n): """ The n-th composite number. This function returns the n-th composite number, starting with 4. ==input== n -- integer >=1 . ==output== integer -- composite(n) ==examples== >>>oeis_core.A002808(1) 4 >>>oeis_core.A002808(2) 6 >>>oeis_core.A002808(3) 8 ==author== Richard J. Mathar 2010-07-01 """ c = 3 i = 0 while i < n: c += 1 while prime.primeq(c): c += 1 i += 1 return c def A002808_list(n): """ The first n composite numbers. This function returns a list of the first n composite numbers, starting with 4. ==input== n -- integer >=1 . ==output== list -- Containing n elements, starting with 4, 6,8 ,9, 10. ==examples== >>>oeis_core.A002808_list(0) [] >>>oeis_core.A002808_list(10) [4, 6, 8, 9, 10, 12, 14, 15, 16, 18 ==author== Richard J. Mathar 2010-07-01 """ c = 3 a = [] while len(a) < n: while prime.primeq(c): c += 1 a.append(c) c += 1 return a def isA002808(n): """ Test the argument for compositeness. This function tests the number against being a member of the A002808 sequence, so for all negative arguments and for all arguments <4 the result will be False, although numbers like -28= 4*(-7) are composite. ==input== n -- integer . ==output== True or False ==examples== >>>oeis_core.isA002808(-12) False >>>oeis_core.isA002808(-11) False >>>oeis_core.isA002808(0) False >>>oeis_core.isA002808(1) False >>>oeis_core.isA002808(3) False >>>oeis_core.isA002808(4) True >>>oeis_core.isA002808(888) True ==author== Richard J. Mathar 2010-07-01 """ return ( n >= 4 and (not prime.primeq(n) ) ) def A003418(n): """ The least common multiple of the numbers 1 to n. ==input== n -- integer >=0. ==output== integer -- 1 if n =0, else lcm(1,2,3,4,....n) ==examples== >>>oeis_core.A003418(12) 27720 ==author== Richard J. Mathar 2010-07-03 """ if n < 0: raise ValueError('non-positive argument '+str(n)) a = 1 for i in range(2,n+1): a = gcd.lcm(a,i) return a def A003484(n): """ Radon function. ==input== n -- integer >0. ==output== integer -- If n = 2**(4*b+c)*d with d odd, then the value is 8*b+2**c ==examples== >>>oeis_core.A003484(1) 1 >>>oeis_core.A003484(2) 2 >>>oeis_core.A003484(16) 9 ==author== Richard J. Mathar 2010-09-06 """ if n <= 0: raise ValueError('non-positive argument '+str(n)) p2 = A007814(n) b, c = divmod(p2,4) return 8*b + 2**c def A004011(n): """ Theta series of the D_4 lattice. ==input== n -- integer >=0. ==output== integer -- function value ==examples== >>>oeis_core.A004011(4) 24 >>>oeis_core.A004011(5) 144 ==author== Richard J. Mathar 2010-09-06 """ if n < 0: raise ValueError('negative argument '+str(n)) if n == 0: return 1 else: return 24*A000593(n) def A004011_list(n): """ The first n values of the theta series of the D_4 lattice. ==input== n -- integer >=0. ==output== list -- The first n values of [1, 24, 24, 96, 24, 144, 96,..] ==examples== >>>oeis_core.A004011_list(4) [1, 24, 24, 96] ==author== Richard J. Mathar 2010-09-06 """ if n < 0: raise ValueError('negative argument '+str(n)) if n == 0: return [] L = [1] if n == 1: return L else: L.extend([24*A000593(i) for i in range(1,n)]) return L def A004526(n): """ floor(n/2). ==input== n -- integer >=0. ==output== integer -- half of n, rounded down. ==examples== >>>oeis_core.A004526(11) 5 ==author== Richard J. Mathar 2010-07-03 """ if n < 0: raise ValueError('non-positive argument '+str(n)) return n//2 def A004526_list(n): """ ==examples== >>>oeis_core.A004526_list(11) [0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5] ==author== Richard J. Mathar 2010-07-03 """ return [i//2 for i in range(n)] def A005117(n): """ The n-th squarefree number, starting at 1. ==input== n -- integer > 0 . ==output== integer -- the n-th squarefree number. This equals n if 1<=n<=3. ==examples== >>>oeis_core.A005117(1) 1 >>>oeis_core.A005117(7) 10 ==author== Richard J. Mathar 2010-07-02 """ if n <=0: raise ValueError('non-positive argument '+str(n)) c = 0 i = 0 while c< n: i += 1 c += 1 while not isA005117(i): i += 1 return i def A005117_list(n): """ List of the first n squarefree numbers. Return the squarefree numbers [1, 2, 3, 5, 6, 7, 10, 11, 13, 14..]. ==input== n -- integer > 0 . ==output== list -- of length n with the lowest squarefree numbers. ==examples== >>>oeis_core.isA005117_list(1) [1] >>>oeis_core.A005117_list(10) [1, 2, 3, 5, 6, 7, 10, 11, 13, 14] ==author== Richard J. Mathar 2010-07-02 """ a = [] i = 0 while len(a) < n: i += 1 a.append(i) while not isA005117(i): i += 1 return a def isA005117(n): """ Test for being squarefree. Returns True the argument is a squarefree number (product of distinct primes) >=1. ==input== n -- integer > 0 . ==output== boolean -- True if the argument is squarefree. ==examples== >>>oeis_core.isA005117(1) True >>>oeis_core.isA005117(10) True >>>oeis_core.isA005117(11) True >>>oeis_core.isA005117(12) False ==author== Richard J. Mathar 2010-07-02 """ if n < 1: return False else: f = factor.misc.FactoredInteger(n) for e in f.factors.values(): if e >1: return False return True def A005230(n): """ Stern's sequence: a(1)=1, otherwise the sum of the previous m terms, where m=ceil((sqrt(8*n+1)-1)/2) . ==input== n -- integer ==output== integer -- The n'th value in [1,1,2,3,6,11,20,40,77] ==examples== >>>oeis_core.A005230(1) 1 >>>oeis_core.A005230(14) 2200 ==author== Richard J. Mathar 2010-09-06 """ return A005230_list(n)[-1] def A005230_list(n): """ The first n terms of Stern's sequence. ==input== n -- integer ==output== list -- The first n values of [1,1,2,3,6,11,20,40,77,..] ==examples== >>>oeis_core.A005230_list(1) [1] >>>oeis_core.A005230_list(14) [1, 1, 2, 3, 6, 11, 20, 40, 77, 148, 285, 570, 1120, 2200] ==author== Richard J. Mathar 2010-09-06 """ a = [1, 1, 2, 3] while len(a) < n: i = len(a) m = oeis_bulk.A002024(i) s = 0 for j in range(1,m+1): s += a[-j] a.append(s) return a[0:n] def A005408(n): """ 2n+1 . ==input== n -- integer ==output== integer -- 2*n+1 ==examples== >>>oeis_core.A005408(0) 1 >>>oeis_core.A005408(2) 5 ==author== Richard J. Mathar 2010-07-01 """ return 2*n+1 def A005408_list(n): """ The first n odd numbers. This function returns a list of the first n odd numbers ==input== n -- integer >=1 . ==output== list -- Containing n elements, starting with 1,3,5,7,.. ==examples== >>>oeis_core.A005408_list(0) [] >>>oeis_core.A005408_list(3) [1, 3, 5] ==author== Richard J. Mathar 2010-07-01 """ return [2*i+1 for i in range(n)] def isA005408(n): """ Test the argument for being a positive odd number. ==input== n -- integer . ==output== boolean -- True or False ==examples== >>>oeis_core.isA005408(-11) False >>>oeis_core.isA005408(11) True >>>oeis_core.isA005408(10) False ==author== Richard J. Mathar 2010-07-01 """ return ( n >= 1 and ( (n & 1) == 1) ) def A005843(n): """ 2n . Note that there are easier programs to multiply a number by 2 than calling this function :-) ==input== n -- integer ==output== integer -- 2*n ==examples== >>>oeis_core.A005843(0) 0 >>>oeis_core.A005843(2) 4 ==author== Richard J. Mathar 2010-07-02 """ return 2*n def A005843_list(n): """ The first n even numbers. This function returns a list of the first n even numbers, starting at 0. ==input== n -- integer >=1 . ==output== list -- Containing n elements, starting with 0,2,4,6,.. ==examples== >>>oeis_core.A005843_list(0) [] >>>oeis_core.A005843_list(3) [0, 2, 4] ==author== Richard J. Mathar 2010-07-02 """ return [2*i for i in range(n)] def A006530(n): """ The largest prime divisor. This function returns the largest prime dividing n. ==input== n -- integer >0 ==output== integer -- the largest prime factor, or 1 if n==1. ==examples== >>>oeis_core.A006530(1) 1 >>>oeis_core.A006530(3) 3 >>>oeis_core.A006530(4) 2 ==author== Richard J. Mathar 2010-07-01 """ if n==1: return 1 f = factor.misc.FactoredInteger(n) return max(f.prime_divisors()) def A006530_list(n): """ List of largest prime factors from 1 to n. ==input== n -- integer >=1 . ==output== list -- the first n elements of A006530. ==examples== >>>oeis_core.A006530_list(10) [1, 2, 3, 2, 5, 3, 7, 2, 3, 5] ==author== Richard J. Mathar 2010-07-01 """ return [A006530(i) for i in range(1,n+1)] def A006894(n): """ Number of planted 3-trees of height <n. ==input== n -- integer > 0 ==output== list -- Recursively a(n+1)= a(n)*(a(n)+1)/2, starting a(1)=1. ==examples== >>>oeis_core.A006894(1) 1 >>>oeis_core.A006894(2) 2 >>>oeis_core.A006894(3) 4 ==author== Richard J. Mathar 2010-09-06 """ if n <= 0: raise ValueError('non-positive argument '+str(n)) a = 1 for i in range(n-1): a = a*(1+a)//2 a += 1 return a def A006894_list(n): """ Recursively a(n+1) = 1+a(n)*(a(n)+1)/2. ==input== n -- integer >= 0 ==output== list -- The first n elements of [1, 2, 4, 11, 67..] ==examples== >>>oeis_core.A006894_list(7) [1, 2, 4, 11, 67, 2279, 2598061] ==author== Richard J. Mathar 2010-09-06 """ a = [1,2] while len(a) < n: s = 1+a[-1]*(1+a[-1])//2 a.append(s) return a[0:n] def A007318(n,k): """ This function returns binomial(n,k) =C(n,k) ==input== n,k -- positive integers n>=0 ==output== integer -- function value ==examples== >>>oeis_core.A007318(-3,4) ValueError: non-positive number: -3 >>>oeis_core.A007318(4,2) 6 ==author== Richard J. Mathar 2010-06-29 """ return combinatorial.binomial(n,k) def A008275(n,k): """ This function returns Stirling1(n,k) ==input== n,k -- positive integers n>=1 ==output== integer -- function value ==examples== >>>oeis_core.A008275(-3,4) 0 >>>oeis_core.A008275(5,3) 35 >>>oeis_core.A008275(5,2) -50 ==author== Richard J. Mathar 2010-06-29 """ return combinatorial.stirling1(n,k) def A008275_row(n): """ Row of stirling1(n,.) numbers. ==input== n -- positive integer n>=1 ==output== list -- Stirling1(n,1) up to Stirling1(n,n) ==examples== >>>oeis_core.A008275_row(3) [2, -3, 1] ==author== Richard J. Mathar 2010-07-03 """ if n <= 0: raise ValueError('non-positive argument '+str(n)) return [A008275(n,k) for k in range(1,n+1)] def A008277(n,k): """ Stirling number of the second kind. This function returns Stirling2(n,k) ==input== n,k -- positive integers n>=1 ==output== integer -- function value ==examples== >>>oeis_core.A008277(-3,4) 0 >>>oeis_core.A008277(5,3) 25 >>>oeis_core.A008277(5,2) 15 ==author== Richard J. Mathar 2010-06-29 """ return combinatorial.stirling2(n,k) def A008292(n,k): """ Eulerian(n,k). This function returns the eulerian number in row n, column k. ==input== n,k -- positive integers n>=1, 1<=k<=n ==output== integer -- function value ==examples== >>> oeis_core.A008292(1,1) 1 >>> oeis_core.A008292(3,2) 4 ==author== Richard J. Mathar 2010-07-03 """ if n < 1: return 0 if k < 1 or k>n: return 0 a = 0 for j in range(k+1): a += (-1)**j * combinatorial.binomial(n+1,j)* ((k-j)**n) return a def A008292_row(n): """ List of Eulerian(n,k) for 1<=k<=n. This function returns the eulerian number in row n. ==input== n -- integer n>=1. ==output== list -- the values Eulerian(n,.) ==examples== >>> oeis_core.A008292_row(3) [1, 4, 1] >>> oeis_core.A008292_row(10) [1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1] ==author== Richard J. Mathar 2010-07-03 """ if n < 1: raise ValueError('non-positive argument '+str(n)) return [A008292(n,k) for k in range(1,n+1)] def A008683(n): """ This function returns mu(n), the Moebius function ==input== n -- positive integer ==output== integer -- function value ==examples== >>>oeis_core.A008683(11) -1 ==author== Richard J. Mathar 2010-06-29 """ return multiplicative.moebius(n) def A008683_list(n): """ This function returns the first n elements of the Moebius function ==input== n -- positive integer ==output== list -- elements mu(1) through mu(n) ==examples== >>>oeis_core.A008683_list(10) [1, -1, -1, 0, -1, 1, -1, 0, 0, 1] ==author== Richard J. Mathar 2010-06-29 """ return [A008683(i) for i in range(1,n+1)] def A018252(n): """ The n-th positive non-prime number. Starting with 1 and4 for argument n=1 and 2. ==input== n -- integer >0. ==output== integer -- The corresponding element of [1, 4, 6, 8, 9, 10, 12, 14, 15..] ==examples== >>>oeis_core.A018252(1) 1 >>>oeis_core.A018252(2) 4 >>>oeis_core.A018252(3) 6 ==author== Richard J. Mathar 2010-07-03 """ if n <= 0: raise ValueError('non-positive argument '+str(n)) if n == 1: return 1 return A002808(n-1) def A027641(n): """ This function returns the numerator of the Bernoulli number (n) ==input== n -- positive integer ==output== integer -- function value ==examples== >>>oeis_core.A027641(6) 1 >>>oeis_core.A027641(12) -691 ==author== Richard J. Mathar 2010-06-29 """ return combinatorial.bernoulli(n).numerator def A027642(n): """ This function returns the denominator of the Bernoulli number (n) ==input== n -- positive integer ==output== integer -- function value ==examples== >>>oeis_core.A027642(6) 42 ==author== Richard J. Mathar 2010-06-29 """ return combinatorial.bernoulli(n).denominator def A049310(n,k): """ Coefficients of Chebyshev U(n,x/2) polynomial. This function returns the coefficient [x**k] S(n,x). ==input== n -- integer >=0 k -- integer 0<=k<=n ==output== integer -- function value ==examples== >>>oeis_core.A049310(3,1) -2 >>>oeis_core.A049310(7,3) 10 ==author== Richard J. Mathar 2010-07-03 """ if k>n or k<0 or n<0: return 0 if (n+k) %2 ==1: return 0 nkhalf = (n+k)//2 return (-1)**(nkhalf+k) * combinatorial.binomial(nkhalf,k) def A049310_row(n): """ Coefficients of Chebyshev U(n,x/2) polynomial. This function returns the the cofficients of x**0, x**1 etc as a list ==input== n -- integer >=0 ==output== list -- the n+1 integer coefficients of the polynomial ==examples== >>>oeis_core.A049310_row(3) [0, -2, 0, 1] >>>oeis_core.A049310_row(7) [0, -4, 0, 10, 0, -6, 0, 1] ==author== Richard J. Mathar 2010-07-03 """ if n<0: return [] return [ A049310(n,k) for k in range(n+1)]
