Displaying 1-10 of 111 results found.
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a(n) = n! + 1.
(Formerly N0107)
+10
91
2, 2, 3, 7, 25, 121, 721, 5041, 40321, 362881, 3628801, 39916801, 479001601, 6227020801, 87178291201, 1307674368001, 20922789888001, 355687428096001, 6402373705728001, 121645100408832001
COMMENTS
"For n = 4, 5 and 7, n!+1 is a square. Sierpiński asked if there are any other values of n with this property." p. 82 of Ogilvy and Anderson (see A146968). Number of {12,12*,1*2,21*,2*1}-avoiding signed permutations in the hyperoctahedral group.
After Wilson's Theorem: if (n+1) is prime then (n+1) is the smallest prime factor of a(n). - Karl-Heinz Hofmann, Aug 21 2024
REFERENCES
C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, 1966, p. 82.
Wacław Sierpiński, On some unsolved problems of arithmetics, Scripta Mathematica, vol. 25 (1960), p. 125.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
LINKS
Arthur T. White, Ringing the changes, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 2, 203-215.
FORMULA
0 = a(n)*(a(n+1) - 5*a(n+2) + 5*a(n+3) - a(n+4)) + a(n+1)*(a(n+1) + a(n+2) - 6*a(n+3) + 2*a(n+4)) + a(n+2)*(3*a(n+2) - a(n+3) - a(n+4)) + a(n+3)*(a(n+3)) if n>=0. - Michael Somos, Apr 23 2014 E.g.f: exp(x) + 1/(1 - x).
EXAMPLE
G.f. = 2 + 2*x + 3*x^2 + 7*x^3 + 25*x^4 + 121*x^5 + 721*x^6 + 5041*x^7 + ...
PROG
(Haskell)
a038507 = (+ 1) . a000142
a038507_list = 2 : f 1 2 where
f x y = z : f (x + 1) z where z = x * (y - 1) + 1
(Python)
from math import factorial
Numbers k such that k! - 1 is prime.
(Formerly M2321)
+10
87
3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, 103040, 147855, 208003
COMMENTS
The corresponding primes n!-1 are often called factorial primes.
REFERENCES
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 166, p. 53, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, Section A2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 719 at p. 160.
LINKS
Eric Weisstein's World of Mathematics, Factorial.
EXAMPLE
The sequence of numbers n! - 1 together with their prime indices begins:
1: {}
5: {3}
23: {9}
119: {4,7}
719: {128}
5039: {675}
40319: {9,273}
362879: {5,5,430}
3628799: {10,11746}
39916799: {6,7,9,992}
479001599: {25306287}
6227020799: {270,256263}
87178291199: {3610490805}
1307674367999: {7,11,11,16,114905}
20922789887999: {436,318519035}
355687428095999: {8,21,10165484947}
6402373705727999: {17,20157,25293727}
121645100408831999: {119,175195,4567455}
2432902008176639999: {11715,659539127675}
(End)
PROG
(Magma) [n: n in [0..500] | IsPrime(Factorial(n)-1)]; // Vincenzo Librandi, Sep 07 2017 (Python)
from sympy import factorial, isprime
A002982_list = [n for n in range(1, 10**2) if isprime(factorial(n)-1)] # Chai Wah Wu, Apr 04 2021
CROSSREFS
Cf. A002981 (numbers n such that n!+1 is prime).
KEYWORD
hard,more,nonn,nice,changed
EXTENSIONS
21480 sent in by Ken Davis (ken.davis(AT)softwareag.com), Oct 29 2001
Updated Feb 26 2007 by Max Alekseyev, based on progress reported in the Carmody web site. Inserted missing 21480 and 34790 (see Caldwell). Added 94550, discovered Oct 05 2010. Eric W. Weisstein, Oct 06 2010 103040 was discovered by James Winskill, Dec 14 2010. It has 471794 digits. Corrected by Jens Kruse Andersen, Mar 22 2011
Hyperfactorials: Product_{k = 1..n} k^k.
(Formerly M3706 N1514)
+10
82
1, 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, 55696437941726556979200000, 21577941222941856209168026828800000, 215779412229418562091680268288000000000000000, 61564384586635053951550731889313964883968000000000000000
COMMENTS
A054374 gives the discriminants of the Hermite polynomials in the conventional (physicists') normalization, and A002109 (this sequence) gives the discriminants of the Hermite polynomials in the (in my opinion more natural) probabilists' normalization. See refs Wikipedia and Szego, eq. (6.71.7). - Alan Sokal, Mar 02 2012
a(n) = (-1)^n/det(M_n) where M_n is the n X n matrix m(i,j) = (-1)^i/i^j. - Benoit Cloitre, May 28 2002 a(n) = determinant of the n X n matrix M(n) where m(i,j) = B(n,i,j) and B(n,i,x) denote the Bernstein polynomial: B(n,i,x) = binomial(n,i)*(1-x)^(n-i)*x^i. - Benoit Cloitre, Feb 02 2003 Number of trailing zeros ( A246839) increases every 5 terms since the exponent of the factor 5 increases every 5 terms and the exponent of the factor 2 increases every 2 terms. - Chai Wah Wu, Sep 03 2014 Also the number of minimum distinguishing labelings in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017 Also shows up in a term in the solution to the generalized version of Raabe's integral. - Jibran Iqbal Shah, Apr 24 2021
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 135-145.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 477.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
G. Szego, Orthogonal Polynomials, American Mathematical Society, 1981 edition, 432 Pages.
LINKS
Eric Weisstein's World of Mathematics, K-Function.
FORMULA
Determinant of n X n matrix m(i, j) = binomial(i*j, i). - Benoit Cloitre, Aug 27 2003 a(n) = exp(zeta'(-1, n + 1) - zeta'(-1)) where zeta(s, z) is the Hurwitz zeta function. - Peter Luschny, Jun 23 2012 G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1 + k^k*x). - Paul D. Hanna, Oct 02 2013 a(n) ~ A * n^(n*(n+1)/2 + 1/12) / exp(n^2/4), where A = 1.2824271291006226368753425... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Feb 20 2015 a(n) = Product_{k=1..n} ff(n,k) where ff denotes the falling factorial. - Peter Luschny, Nov 29 2015 log a(n) = (1/2) n^2 log n - (1/4) n^2 + (1/2) n log n + (1/12) log n + log(A) + o(1), where log(A) = A225746 is the logarithm of Glaisher's constant. - Charles R Greathouse IV, Mar 27 2020 Sum_{n>=1} (-1)^(n+1)/a(n) = A347352. (End) a(n) = e^(Integral_{x=1..n+1} (x - 1/2 - log(sqrt(2*Pi)) + (n+1-x)*Psi(x)) dx), where Psi(x) is the digamma function.
a(n) = e^(Integral_{x=1..n} (x + 1/2 - log(sqrt(2*Pi)) + log(Gamma(x+1))) dx). (End)
MAPLE
f := proc(n) local k; mul(k^k, k=1..n); end;
A002109 := n -> exp(Zeta(1, -1, n+1)-Zeta(1, -1));
MATHEMATICA
Join[{1}, FoldList[Times, #^#&/@Range[15]]] (* Harvey P. Dale, Nov 02 2023 *)
PROG
(PARI) a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/prod(j=1, k+1, (1+j^j*x+x*O(x^n)) )), n) \\ Paul D. Hanna, Oct 02 2013 (Haskell)
a002109 n = a002109_list !! n
(Python)
for n in range(1, 10):
(Sage)
a = lambda n: prod(falling_factorial(n, k) for k in (1..n))
CROSSREFS
Cf. A000178, A000142, A000312, A001358, A002981, A002982, A100015, A005234, A014545, A018239, A006794, A057704, A057705, A054374. Cf. A074962 [Glaisher-Kinkelin constant, also gives an asymptotic approximation for the hyperfactorials].
Numbers k such that (k! + 3)/3 is prime.
+10
57
3, 5, 6, 8, 11, 17, 23, 36, 77, 93, 94, 109, 304, 497, 1330, 1996, 3027, 3053, 4529, 5841, 20556, 26558, 28167
COMMENTS
a(21) > 20000. The PFGW program has been used to certify all the terms up to a(20), using the "N-1" deterministic test. - Giovanni Resta, Mar 31 2014
PROG
(Magma) [n: n in [0..500] | IsPrime((Factorial(n)+3) div 3)]; // Vincenzo Librandi, Dec 12 2011
CROSSREFS
Cf. n!/m-1 is a prime: A002982, A082671, A139056, A139199- A139205; n!/m+1 is a prime: A002981, A082672, A089085, A139061, A139058, A139063, A139065, A151913, A137390, A139071 (1<=m<=10).
EXTENSIONS
1330 from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008
Numbers n such that (n! + 2)/2 is a prime.
+10
56
2, 4, 5, 7, 8, 13, 16, 30, 43, 49, 91, 119, 213, 1380, 1637, 2258, 4647, 9701, 12258
PROG
(PARI) \\ x such that (x!+2)/2 is prime
xfactpk(n, k=2) = { for(x=2, n, y = (x!+k)/k; if(isprime(y), print1(x, ", ")) ) }
(Magma) [ n: n in [1..300] | IsPrime((Factorial(n)+2) div 2) ];
CROSSREFS
Cf. n!/m-1 is a prime: A002982, A082671, A139056, A139199- A139205; n!/m+1 is a prime: A002981, A082672, A089085, A139061, A139058, A139063, A139065, A151913, A137390, A139071 (1<=m<=10).
EXTENSIONS
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008
Numbers k for which (k!-3)/3 is prime.
+10
56
4, 6, 12, 16, 29, 34, 43, 111, 137, 181, 528, 2685, 39477, 43697
COMMENTS
Corresponding primes (k!-3)/3 are in A139057. a(13) > 10000. The PFGW program has been used to certify all the terms up to a(12), using a deterministic test which exploits the factorization of a(n) + 1. - Giovanni Resta, Mar 28 2014 98166 is a member of the sequence but its index is not yet determined. The interval where sieving and tests were not run is [60000,90000]. - Serge Batalov, Feb 24 2015
MATHEMATICA
a = {}; Do[If[PrimeQ[(-3 + n!)/3], AppendTo[a, n]], {n, 1, 1000}]; a
PROG
(PARI) for(n=1, 1000, if(floor(n!/3-1)==n!/3-1, if(ispseudoprime(n!/3-1), print(n)))) \\ Derek Orr, Mar 28 2014
EXTENSIONS
Definition corrected by Derek Orr, Mar 28 2014
Primes of the form n!! - 1.
+10
51
2, 7, 47, 383, 10321919, 51011754393599, 1130138339199322632554990773529330319359999999, 73562883979319395645666688474019139929848516028923903999999999, 4208832729023498248022825567687608993477547383960134557368319999999999
REFERENCES
G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 158.
EXAMPLE
6!! - 1 = 6*4*2 - 1 = 48 - 1 = 47, which is prime.
8!! - 1 = 8*6*4*2 - 1 = 384 - 1 = 383, which is prime.
MAPLE
SFACT:= proc(n) local i, j, k; for k from 1 by 1 to n do i:=k; j:=k-2; while j >0 do i:=i*j; j:=j-2; od: if isprime(i-1) then print(i-1); fi; od: end: SFACT(100);
PROG
(PARI) print1(2); for(n=1, 1e3, if(ispseudoprime(t=n!<<n-1), print1(", "t))) \\ Charles R Greathouse IV, Jun 16 2011
CROSSREFS
Cf. A093173 = primes of the form (2^n * n!) - 1.
Numbers k for which (9 + k!)/9 is prime.
+10
50
8, 46, 87, 168, 259, 262, 292, 329, 446, 1056, 3562, 11819, 26737
COMMENTS
No other k exists, for k <= 6000. - Dimitris Zygiridis (dmzyg70(AT)gmail.com), Jul 25 2008
The next number in the sequence, if one exists, is greater than 10944. - Robert Price, Mar 16 2010 Borrowing from A139074 another term in this sequence is 26737. There may be others between 10944 and 26737. - Robert Price, Dec 13 2011 There are no other terms for k < 26738. - Robert Price, Feb 10 2012
EXAMPLE
a(11) = 3562 because 3562 is the 11th natural number for which k!/9 + 1 is prime. 3562 is the new term.
MATHEMATICA
a = {}; Do[If[PrimeQ[(n! + 9)/9], AppendTo[a, n]], {n, 1, 500}]; a
CROSSREFS
Cf. A139068 (primes of the form (9 + k!)/9).
EXTENSIONS
a(11) from Dimitris Zygiridis (dmzyg70(AT)gmail.com), Jul 25 2008
Numbers n such that (5+n!)/5 is prime.
+10
26
7, 9, 11, 14, 19, 23, 45, 121, 131, 194, 735, 751, 1316, 1372, 2084, 2562, 5678, 5758, 12533, 24222
COMMENTS
For primes of the form (5+n!)/5 see A139059.
MATHEMATICA
a = {}; Do[If[PrimeQ[(n! + 5)/5], AppendTo[a, n]], {n, 1, 751}]; a
PROG
(Magma) [ n: n in [5..734] | IsPrime((Factorial(n)+5) div 5) ];
(PARI) A139058(n) = local(k=(n!+5)\5); if(isprime(k), k, 0); for(n=5, 800, if( A139058(n)>0, print1(n, ", ")))
CROSSREFS
Cf. n!/m-1 is a prime: A002982, A082671, A139056, A139199- A139205; n!/m+1 is a prime: A002981, A082672, A089085, A139061, A139058, A139063, A139065, A151913, A137390, A139071 (1<=m<=10).
Numbers n for which (4+n!)/4 is prime.
+10
25
4, 5, 6, 13, 21, 25, 32, 40, 61, 97, 147, 324, 325, 348, 369, 1290, 1342, 3167, 6612, 8176, 10990
COMMENTS
For primes of the form (4+k!)/4, see A139060.
MATHEMATICA
a = {}; Do[If[PrimeQ[(n! + 4)/4], AppendTo[a, n]], {n, 1, 500}]; a
Select[Range[500], PrimeQ[(4+#!)/4]&] (* Harvey P. Dale, Mar 24 2011 *)
CROSSREFS
Cf. A082672, A089085, A089130, A117141, A007749, A139056, A139057, A139058, A139059, A139060, A139061, A139061, A139062, A139063, A139064, A139065, A139066, A020458, A139068, A137390, A139070, A139071, A139072. Cf. n!/m-1 is a prime: A002982, A082671, A139056, A139199- A139205; n!/m+1 is a prime: A002981, A082672, A089085, A139061, A139058, A139063, A139065, A151913, A137390, A139071 (1<=m<=10).
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