Displaying 1-10 of 20 results found.
Number of partitions of n into distinct squares.
+10
110
1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 2, 2, 0, 0, 2, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 0, 2, 3, 1, 1, 4, 3, 0, 1, 2, 2, 1, 0, 1, 4, 3, 0, 2, 4, 2, 1, 3, 2, 1, 2, 3, 3, 2, 1, 3, 6, 3, 0, 2, 5, 3, 0, 1, 3, 3, 3, 4
COMMENTS
"WEIGH" transform of squares A000290. Number of partitions of n in which each part i has multiplicity i. Example: a(50)=3 because we have [1,2,2,3,3,3,6,6,6,6,6,6], [1,7,7,7,7,7,7,7], and [3,3,3,4,4,4,4,5,5,5,5,5]. - Emeric Deutsch, Jan 26 2016 The Heinz numbers of integer partitions into distinct pairs are given by A324587. - Gus Wiseman, Mar 09 2019 Equivalent to Emeric Deutsch's comment, a(n) is the number of integer partitions of n where the multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are equal to the distinct parts in increasing order. The Heinz numbers of these partitions are given by A109298. For example, the first 30 terms count the following integer partitions: 1: (1)
4: (22)
5: (221)
9: (333)
10: (3331)
13: (33322)
14: (333221)
16: (4444)
17: (44441)
20: (444422)
21: (4444221)
25: (55555)
25: (4444333)
26: (555551)
26: (44443331)
29: (5555522)
29: (444433322)
30: (55555221)
30: (4444333221)
The case where the distinct parts are taken in decreasing order is A324572, with Heinz numbers given by A324571. (End)
FORMULA
G.f.: Product_{n>=1} ( 1+x^(n^2) ).
a(n) ~ exp(3 * 2^(-5/3) * Pi^(1/3) * ((sqrt(2)-1)*zeta(3/2))^(2/3) * n^(1/3)) * ((sqrt(2)-1)*zeta(3/2))^(1/3) / (2^(4/3) * sqrt(3) * Pi^(1/3) * n^(5/6)), where zeta(3/2) = A078434. - Vaclav Kotesovec, Dec 09 2016 See Murthy, Brack, Bhaduri, Bartel (2018) for a more complete asymptotic expansion. - N. J. A. Sloane, Aug 17 2018
EXAMPLE
a(50)=3 because we have [1,4,9,36], [1,49], and [9,16,25]. - Emeric Deutsch, Jan 26 2016 The first 30 terms count the following integer partitions:
1: (1)
4: (4)
5: (4,1)
9: (9)
10: (9,1)
13: (9,4)
14: (9,4,1)
16: (16)
17: (16,1)
20: (16,4)
21: (16,4,1)
25: (25)
25: (16,9)
26: (25,1)
26: (16,9,1)
29: (25,4)
29: (16,9,4)
30: (25,4,1)
30: (16,9,4,1)
(End)
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i-1))))
end:
a:= n-> b(n, isqrt(n)):
MATHEMATICA
nn=10; CoefficientList[Series[Product[(1+x^(k*k)), {k, nn}], {x, 0, nn*nn}], x] (* T. D. Noe, Jul 24 2006 *) b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i^2 > n, 0, b[n - i^2, i-1]]]]; a[n_] := b[n, Floor[Sqrt[n]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 21 2015, after Alois P. Heinz *) nmax = 20; poly = ConstantArray[0, nmax^2 + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += poly[[j - k^2 + 1]], {j, nmax^2, k^2, -1}]; , {k, 2, nmax}]; poly (* Vaclav Kotesovec, Dec 09 2016 *) Table[Length[Select[IntegerPartitions[n], Reverse[Union[#]]==Length/@Split[#]&]], {n, 30}] (* Gus Wiseman, Mar 09 2019 *)
PROG
(PARI) a(n)=polcoeff(prod(k=1, sqrt(n), 1+x^k^2), n)
(Python)
from functools import cache
from sympy.core.power import isqrt
@cache
def b(n, i):
if n == 0: return 1
if i == 0: return 0
i2 = i*i
return b(n, i-1) + (0 if i2 > n else b(n - i2, i-1))
a = lambda n: b(n, isqrt(n))
print([a(n) for n in range(1, 101)]) # Darío Clavijo, Nov 30 2023
CROSSREFS
Cf. A001156 (non-strict case), A001462, A005117, A052335, A078135, A109298, A114638, A117144, A324571, A324572, A324587, A324588.
Triangle read by rows: T(n,k) is the number of partitions of n such that the sum of the parts, counted without multiplicities, is equal to k (n>=1, k>=1).
+10
78
1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 0, 2, 1, 3, 1, 1, 3, 1, 1, 4, 1, 0, 3, 2, 2, 2, 5, 1, 1, 3, 3, 2, 4, 2, 6, 1, 0, 5, 2, 3, 4, 4, 3, 8, 1, 1, 4, 3, 4, 7, 4, 5, 3, 10, 1, 0, 5, 3, 4, 7, 7, 6, 6, 5, 12, 1, 1, 6, 4, 3, 12, 6, 8, 7, 9, 5, 15, 1, 0, 6, 4, 5, 10, 10, 9, 10, 11, 10, 7, 18, 1, 1, 6, 4, 5, 15, 11, 13, 9, 16, 11, 13, 8, 22
COMMENTS
Conjecture: Reverse the rows of the table to get an infinite lower-triangular matrix b with 1's on the main diagonal. The third diagonal of the inverse of b is minus A137719. - George Beck, Oct 26 2019 Proof: The reversed rows yield the matrix I+N where N is strictly lower triangular, N[i,j] = 0 for j >= i, having its 2nd diagonal equal to the 2nd column (1, 0, 1, 0, 1, ...): N[i+1,i] = A000035(i), i >= 1, and 3rd diagonal equal to the 3rd column of this triangle, (2, 1, 2, 3, 3, 3, ...): N[i+2,i] = A137719(i), i >= 1. It is known that (I+N)^-1 = 1 - N + N^2 - N^3 +- .... Here N^2 has not only the second but also the 3rd diagonal zero, because N^2[i+2,i] = N[i+2,i+1]*N[i+1,i] = A000035(i+1)* A000035(i) = 0. Therefore the 3rd diagonal of (I+N)^-1 is equal to - A137719 without leading 0. - M. F. Hasler, Oct 27 2019 Also the number of ways to write n-k as a nonnegative linear combination of a strict integer partition of k. Also the number of ways to write n as a (strictly) positive linear combination of a strict integer partition of k. Row n=7 counts the following:
7*1 . 1*2+5*1 1*3+4*1 1*3+2*2 1*5+2*1 1*7
2*2+3*1 2*3+1*1 1*4+3*1 1*3+1*2+2*1 1*4+1*3
3*2+1*1 1*5+1*2
1*6+1*1
1*4+1*2+1*1
(End)
FORMULA
G.f.: -1 + Product_{j>=1} (1 + t^j*x^j/(1-x^j)).
Sum_{k=1..n} k*T(n,k) = A014153(n-1).
EXAMPLE
T(10,7) = 4 because we have [6,1,1,1,1], [4,3,3], [4,2,2,1,1] and [4,2,1,1,1,1] (6+1=4+3=4+2+1=7).
Triangle starts:
1;
1, 1;
1, 0, 2;
1, 1, 1, 2;
1, 0, 2, 1, 3;
1, 1, 3, 1, 1, 4;
1, 0, 3, 2, 2, 2, 5;
1, 1, 3, 3, 2, 4, 2, 6;
1, 0, 5, 2, 3, 4, 4, 3, 8;
1, 1, 4, 3, 4, 7, 4, 5, 3, 10;
1, 0, 5, 3, 4, 7, 7, 6, 6, 5, 12;
1, 1, 6, 4, 3, 12, 6, 8, 7, 9, 5, 15;
...
MAPLE
g:= -1+product(1+t^j*x^j/(1-x^j), j=1..40): gser:= simplify(series(g, x=0, 18)): for n from 1 to 14 do P[n]:=sort(coeff(gser, x^n)) od: for n from 1 to 14 do seq(coeff(P[n], t^j), j=1..n) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; local f, g, j;
if n=0 then [1] elif i<1 then [ ] else f:= b(n, i-1);
for j to n/i do
f:= zip((x, y)->x+y, f, [0$i, b(n-i*j, i-1)[]], 0)
od; f
fi
end:
T:= n-> subsop(1=NULL, b(n, n))[]:
MATHEMATICA
max = 14; s = Series[-1+Product[1+t^j*x^j/(1-x^j), {j, 1, max}], {x, 0, max}, {t, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {t, 0, k}]; Table[t[n, k], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 17 2014 *) Table[Length[Select[IntegerPartitions[n], Total[Union[#]]==k&]], {n, 0, 10}, {k, 0, n}] (* Gus Wiseman, Aug 29 2023 *)
PROG
(PARI) A116861(n, k, s=0)={forpart(X=n, vecsum(Set(X))==k&&s++, k); s} \\ M. F. Hasler, Oct 27 2019
CROSSREFS
Cf. A114638 (count partitions with #parts = sum(distinct parts)). For subsets instead of partitions we have A026820. This statistic is ranked by A066328. Partial sums of columns are columns of A364911. Same as A364916 (offset 0) with rows reversed. A364912 counts linear combinations of partitions.
Number of integer partitions of n whose smallest part is equal to the number of parts.
(Formerly M0260)
+10
67
1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 23, 25, 29, 33, 38, 42, 49, 54, 62, 69, 78, 87, 99, 109, 123, 137, 154, 170, 191, 211, 236, 261, 290, 320, 357, 392, 435, 479, 530, 582, 644, 706, 779, 854, 940, 1029, 1133, 1237, 1358, 1485
COMMENTS
Or, number of partitions of n in which number of largest parts is equal to the largest part.
a(n) is the number of partitions of n-1 without parts that differ by less than 2 and which have no parts less than three. [MacMahon]
There are two conflicting choices for the offset in this sequence. For the definition given here the offset is 1, and that is what we shall adopt. On the other hand, if one arrives at this sequence via the Rogers-Ramanujan identities (see the next comment), the natural offset is 0.
Related to Rogers-Ramanujan identities: Let G[1](q) and G[2](q) be the generating functions for the two Rogers-Ramanujan identities of A003114 and A003106, starting with the constant term 1. The g.f. for the present sequence is G[3](q) = (G[1](q) - G[2](q))/q = 1+q^3+q^4+q^5+q^6+q^7+2*q^8+2*q^9+3*q^10+.... - Joerg Arndt, Oct 08 2012; N. J. A. Sloane, Nov 18 2015 For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[3](x). - N. J. A. Sloane, Nov 22 2015 From Hardy (H) p. 94, eq. (6.12.1) and Hardy-Wright (H-W), p. 293, eq. (19.14.3) for H_2(a,x) - H_1(a,x) = a*H_1(a*x,x) one finds from the result for H_1(a,x) (in (H) on top on p. 95), after putting a=x, the o.g.f. of a(n) = A003114(n) - A003106(n), n >= 0, with a(0) = 0 as Sum_{m>=0} x^((m+1)^2) / Product_{j=1..m} (1 - x^j). The m=0 term is 1*x^1. See the formula given by Joerg Arndt, Jan 29 2011. This formula has a combinatorial interpretation (found similar to the one given in (H) section 6.0, pp. 91-92 or (H-W) pp. 290-291): a(n) is the number of partitions of n with parts differing by at least 2 and part 1 present. See the example for a(15) below. (End)
The Heinz numbers of these integer partitions are given by A324522. - Gus Wiseman, Mar 09 2019
REFERENCES
G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 92-95.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, pp. 292-294.
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 45, Section 293.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
G.f.: Sum_{m>=1} (x^(m^2)-x^(m*(m+1))) / Product_{i=1..m} (1-x^i) .
G.f.: Sum_{n>=1} x^(n^2)/Product_{k=1..n-1} (1-x^k). - Joerg Arndt, Jan 29 2011 Plouffe in his 1992 dissertation conjectured that this has g.f. = (1+z+z^4+2*z^5-z^3-z^8+3*z^10-z^7+z^9)/(1+z-z^4-2*z^3-z^8+z^10), but Michael Somos pointed out on Jan 22 2008 that this is false. Expansion of ( f(-x^2, -x^3) - f(-x, -x^4) ) / f(-x) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Jan 22 2007 a(n) ~ sqrt(1/sqrt(5) - 2/5) * exp(2*Pi*sqrt(n/15)) / (2*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Nov 01 2016
EXAMPLE
G.f. = x + x^4 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 2*x^10 + 3*x^11 + 3*x^12 + ...
a(15) = 5 because the partitions of 15 where the smallest part equals the number of parts are
3 + 6 + 6,
3 + 5 + 7,
3 + 4 + 8,
3 + 3 + 9, and
2 + 13.
a(15) = 5 because the partitions of 15 with parts differing by at least 2 and part 1 present are: [14,1] obtained from the partition of 11 with one part, [11], added to the first part of the special partition [3,1] of 4 and [11,3,1], [10,4,1], [9,5,1], [8,6,1] from adding all partition of 15 - 9 = 6 with one part, [6], and those with two parts, [5,1], [4,1], [3,3], to the special partition [5,3,1] of 9. - Wolfdieter Lang, Oct 31 2016 a(15) = 5 because the partitions of 14 with parts >= 3 and parts differing by at least 2 are [14], [11,3], [10,4], [9,5] and [8,6]. See the second [MacMahon] comment. This follows from the g.f. G[3](q) given in Andrews - Baxter, eq. (5.1) for i=3, (using summation index m) and m*(m+2) = 3 + 5 + ... + (2*m+1). - Wolfdieter Lang, Nov 02 2016 The a(8) = 1 through a(15) = 5 integer partitions:
(6,2) (7,2) (8,2) (9,2) (10,2) (11,2) (12,2) (13,2)
(3,3,3) (4,3,3) (4,4,3) (5,4,3) (5,5,3) (6,5,3) (6,6,3)
(5,3,3) (6,3,3) (6,4,3) (7,4,3) (7,5,3)
(7,3,3) (8,3,3) (8,4,3)
(9,3,3)
(End)
MAPLE
b:= proc(n, i) option remember; `if`(n<0, 0, `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))))
end:
a:= n-> add(b(n-j^2, j-1), j=0..isqrt(n)):
MATHEMATICA
b[n_, i_] := b[n, i] = If[n<0, 0, If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]]; a[n_] := Sum[b[n-j^2, j-1], {j, 0, Sqrt[n]}]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *) Table[Length[Select[IntegerPartitions[n], Min[#]==Length[#]&]], {n, 30}] (* Gus Wiseman, Mar 09 2019 *)
PROG
(PARI) {a(n) = if( n<1, 0, polcoeff( sum(k=1, sqrtint(n), x^k^2 / prod(j=1, k-1, 1 - x^j, 1 + O(x ^ (n - k^2 + 1) ))), n))} /* Michael Somos, Jan 22 2008 */
CROSSREFS
For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G[0] = G[1]+G[2] is given by A003113. A003106 counts partitions with minimum > length.
A003114 counts partitions with minimum >= length.
A026794 counts partitions by minimum.
A039899 counts partitions with minimum < length.
A039900 counts partitions with minimum <= length.
A239950 counts partitions with minimum equal to number of distinct parts.
Sequences related to balance:
- A010054 counts balanced strict partitions. - A047993 counts balanced partitions. - A098124 counts balanced compositions. - A106529 ranks balanced partitions. - A340596 counts co-balanced factorizations. - A340598 counts balanced set partitions. - A340599 counts alt-balanced factorizations. - A340600 counts unlabeled balanced multiset partitions. - A340653 counts balanced factorizations.
EXTENSIONS
More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 20 2000
a(n) = sum of indices of distinct prime factors of n; here, index(i-th prime) = i.
+10
58
0, 1, 2, 1, 3, 3, 4, 1, 2, 4, 5, 3, 6, 5, 5, 1, 7, 3, 8, 4, 6, 6, 9, 3, 3, 7, 2, 5, 10, 6, 11, 1, 7, 8, 7, 3, 12, 9, 8, 4, 13, 7, 14, 6, 5, 10, 15, 3, 4, 4, 9, 7, 16, 3, 8, 5, 10, 11, 17, 6, 18, 12, 6, 1, 9, 8, 19, 8, 11, 8, 20, 3, 21, 13, 5, 9, 9, 9, 22, 4, 2, 14, 23, 7, 10, 15, 12, 6, 24, 6, 10
COMMENTS
a(n) = the sum of the distinct parts of the partition with Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} (p_j-th prime) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(75) = 5; indeed, the partition having Heinz number 75 = 3*5*5 is [2,3,3] and 2 + 3 = 5. - Emeric Deutsch, Jun 04 2015
FORMULA
G.f.: Sum_{k>=1} k*x^prime(k)/(1-x^prime(k)). - Vladeta Jovovic, Aug 11 2004 Additive with a(p^e) = PrimePi(p), where PrimePi(n) = A000720(n).
EXAMPLE
a(24) = 1 + 2 = 3 because 24 = 2^3 * 3 = p(1)^3 * p(2), p(k) being the k-th prime.
The distinct prime indices of 1..20 and their sums.
1: () = 0
2: (1) = 1
3: (2) = 2
4: (1) = 1
5: (3) = 3
6: (1+2) = 3
7: (4) = 4
8: (1) = 1
9: (2) = 2
10: (1+3) = 4
11: (5) = 5
12: (1+2) = 3
13: (6) = 6
14: (1+4) = 5
15: (2+3) = 5
16: (1) = 1
17: (7) = 7
18: (1+2) = 3
19: (8) = 8
20: (1+3) = 4
(End)
MAPLE
with(numtheory): seq(add(pi(d), d in factorset(n)), n=1..100); # Ridouane Oudra, Aug 19 2019
MATHEMATICA
PrimeFactors[n_Integer] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]; f[n_] := (Plus @@ PrimePi[ PrimeFactors[n]]); Table[ f[n], {n, 91}] (* Robert G. Wilson v, May 04 2004 *)
PROG
(PARI) { for (n=1, 1000, f=factor(n); a=0; for (i=1, matsize(f)[1], a+=primepi(f[i, 1])); write("b066328.txt", n, " ", a) ) } \\ Harry J. Smith, Feb 10 2010 (Python)
from sympy import primepi, primefactors
Partitions of n in which each part k occurs at least k times.
+10
22
1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 8, 9, 10, 12, 15, 16, 19, 21, 25, 28, 32, 34, 41, 46, 51, 55, 64, 70, 79, 86, 97, 106, 119, 129, 146, 159, 175, 190, 214, 232, 256, 277, 306, 334, 367, 394, 434, 472, 515, 556, 607, 654, 714, 770, 836, 901, 978, 1048, 1140, 1226, 1322
COMMENTS
The Heinz numbers of these integer partitions are given by A324525. - Gus Wiseman, Mar 09 2019
FORMULA
G.f.: Product_{k>=1} (1-x^k+x^(k^2))/(1-x^k).
EXAMPLE
a(9)=5 because we have [3,3,3], [2,2,2,2,1], [2,2,2,1,1,1], [2,2,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1].
The a(1) = 1 through a(9) = 5 integer partitions:
1 11 111 22 221 222 2221 2222 333
1111 11111 2211 22111 22211 22221
111111 1111111 221111 222111
11111111 2211111
111111111
(End)
MAPLE
g:=product((1-x^k+x^(k^2))/(1-x^k), k=1..100): gser:=series(g, x=0, 70): seq(coeff(gser, x, n), n=0..66);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1) +add(b(n-i*j, i-1), j=i..n/i)))
end:
a:= n-> b(n$2):
MATHEMATICA
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-1], {j, i, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *) Table[Length[Select[IntegerPartitions[n], And@@Table[Count[#, i]>=i, {i, Union[#]}]&]], {n, 0, 30}] (* Gus Wiseman, Mar 09 2019 *) nmax = 100; CoefficientList[Series[Product[(1-x^k+x^(k^2))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 28 2024 *)
CROSSREFS
Cf. A001462, A003114, A006141, A033461, A039900, A047993, A052335, A064174, A090858, A114638, A115584, A276078, A280204.
Number of partitions of n for which 2*(number of distinct parts) <= (number of parts).
+10
19
1, 0, 1, 1, 2, 2, 6, 6, 10, 13, 20, 26, 39, 50, 71, 87, 121, 156, 208, 265, 348, 440, 566, 712, 906, 1131, 1424, 1766, 2224, 2738, 3390, 4168, 5130, 6266, 7664, 9312, 11332, 13723, 16603, 20004, 24112, 28942, 34708, 41522, 49612, 59031, 70308, 83479, 98992
COMMENTS
Also the number of integer partitions of n whose median difference is 0. For example, the partition (2,2,2,1,1) is counted because its multiset of differences {0,0,0,1} has median 0. - Gus Wiseman, Mar 18 2023
EXAMPLE
Among the 22 partitions of 8, these qualify: [5,1,1,1], [4,4], [4,1,1,1,1], [3,3,1,1], [3,1,1,1,1,1], [2,2,2,2], [2,2,2,1,1], [2,2,1,1,1,1], [2,1,1,1,1,1,1], [1,1,1,1,1,1,1,1], and the remaining 12 do not, so that a(8) = 10.
MATHEMATICA
z = 50; t = Map[Length[Select[IntegerPartitions[#], 2*Length[DeleteDuplicates[#]] <= Length[#] &]] &, Range[z]] (* A237363*) Table[Length[Select[IntegerPartitions[n], Median[Differences[#]]==0&]], {n, 0, 30}] (* Gus Wiseman, Mar 18 2023 *)
CROSSREFS
These partitions have ranks A361204. A116608 counts partitions by number of distinct parts.
Comparing twice the number of distinct parts to the number of parts:
Triangle read by rows where T(n,k) is the number of integer partitions with sum <= n and with distinct parts summing to k.
+10
19
1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 4, 2, 3, 2, 1, 5, 2, 5, 3, 3, 1, 6, 3, 8, 4, 4, 4, 1, 7, 3, 11, 6, 6, 6, 5, 1, 8, 4, 14, 9, 8, 10, 7, 6, 1, 9, 4, 19, 11, 11, 14, 11, 9, 8, 1, 10, 5, 23, 14, 15, 21, 15, 14, 11, 10, 1, 11, 5, 28, 17, 19, 28, 22, 20, 17, 15, 12
COMMENTS
Also the number of ways to write any number up to n as a positive linear combination of a strict integer partition of k.
FORMULA
G.f.: A(x,y) = (1/(1 - x)) * Product_{k>=1} (1 - y^k + y^k/(1 - x^k)). - Andrew Howroyd, Jan 11 2024
EXAMPLE
Triangle begins:
1
1 1
1 2 1
1 3 1 2
1 4 2 3 2
1 5 2 5 3 3
1 6 3 8 4 4 4
1 7 3 11 6 6 6 5
1 8 4 14 9 8 10 7 6
1 9 4 19 11 11 14 11 9 8
1 10 5 23 14 15 21 15 14 11 10
1 11 5 28 17 19 28 22 20 17 15 12
1 12 6 34 21 22 40 28 28 24 24 17 15
1 13 6 40 25 27 50 38 37 34 35 27 22 18
1 14 7 46 29 32 65 49 50 43 51 38 35 26 22
1 15 7 54 33 38 79 62 63 59 68 55 50 41 32 27
Row n = 5 counts the following partitions:
. 1 2 3 4 5
1+1 2+2 1+2 1+3 1+4
1+1+1 1+1+2 1+1+3 2+3
1+1+1+1 1+1+1+2
1+1+1+1+1 1+2+2
Row n = 5 counts the following positive linear combinations:
. 1*1 1*2 1*3 1*4 1*5
2*1 2*2 1*2+1*1 1*3+1*1 1*3+1*2
3*1 1*2+2*1 1*3+2*1 1*4+1*1
4*1 1*2+3*1
5*1 2*2+1*1
MATHEMATICA
Table[Length[Select[Array[IntegerPartitions, n+1, 0, Join], Total[Union[#]]==k&]], {n, 0, 9}, {k, 0, n}]
PROG
(PARI) T(n)={[Vecrev(p) | p<-Vec(prod(k=1, n, 1 - y^k + y^k/(1 - x^k), 1/(1 - x) + O(x*x^n)))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024
CROSSREFS
Columns are partial sums of columns of A116861. Column k = 3 appears to be the partial sums of A137719. A114638 counts partitions where (length) = (sum of distinct parts).
A116608 counts partitions by number of distinct parts.
A364350 counts combination-free strict partitions, complement A364839.
Cf. A002865, A066328, A179009, A236912, A237113, A237667, A364912, A364913, A364915, A364916, A365002, A365004.
Numbers where the sum of distinct prime indices ( A066328) is equal to the number of prime factors counted with multiplicity ( A001222).
+10
13
1, 2, 9, 12, 18, 40, 100, 112, 125, 240, 250, 352, 360, 392, 405, 540, 600, 672, 675, 810, 832, 900, 1008, 1125, 1350, 1372, 1500, 1512, 1701, 1875, 1936, 2112, 2176, 2240, 2250, 2268, 2352, 2401, 3168, 3402, 3528, 3750, 3969, 4752, 4802, 4864, 4992, 5292
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. For example, 540 = prime(1)^2 * prime(2)^3 * prime(3)^1 has sum of distinct prime indices 1 + 2 + 3 = 6, while the number of prime factors counted with multiplicity is 2 + 3 + 1 = 6, so 540 belongs to the sequence. Also Heinz numbers of the integer partitions counted by A114638. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
9: {2,2}
12: {1,1,2}
18: {1,2,2}
40: {1,1,1,3}
100: {1,1,3,3}
112: {1,1,1,1,4}
125: {3,3,3}
240: {1,1,1,1,2,3}
250: {1,3,3,3}
352: {1,1,1,1,1,5}
360: {1,1,1,2,2,3}
392: {1,1,1,4,4}
405: {2,2,2,2,3}
540: {1,1,2,2,2,3}
600: {1,1,1,2,3,3}
672: {1,1,1,1,1,2,4}
MAPLE
with(numtheory):
q:= n-> is(add(pi(p), p=factorset(n))=bigomega(n)):
MATHEMATICA
Select[Range[1000], Total[PrimePi/@First/@FactorInteger[#]]==PrimeOmega[#]&]
Numbers for which the prime indices do not have the same mean as the distinct prime indices.
+10
12
12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189
COMMENTS
First differs from A242416 in having 126. Contains no squarefree numbers or perfect powers.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
EXAMPLE
The terms together with their prime indices begin:
12: {1,1,2}
18: {1,2,2}
20: {1,1,3}
24: {1,1,1,2}
28: {1,1,4}
40: {1,1,1,3}
44: {1,1,5}
45: {2,2,3}
48: {1,1,1,1,2}
50: {1,3,3}
52: {1,1,6}
54: {1,2,2,2}
56: {1,1,1,4}
60: {1,1,2,3}
63: {2,2,4}
68: {1,1,7}
72: {1,1,1,2,2}
The prime indices of 126 are {1,2,2,4} with mean 9/4 and distinct prime indices {1,2,4} with mean 7/3, so 126 is in the sequence.
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Mean[prix[#]]!=Mean[Union[prix[#]]]&]
CROSSREFS
Signature instead of distinct parts: complement A359903, counted by A360068. These partitions are counted by A360242. A316413 = numbers whose prime indices have integer mean, distinct A326621.
Numbers for which the prime indices have the same mean as the distinct prime indices.
+10
12
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 100, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 121, 122, 123, 125, 127, 128, 129, 130
COMMENTS
First differs from A072774 in having 90. First differs from A242414 in lacking 126. Includes all squarefree numbers and perfect powers.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
EXAMPLE
The prime indices of 900 are {3,3,2,2,1,1} with mean 2, and the distinct prime indices are {1,2,3} also with mean 2, so 900 is in the sequence.
MAPLE
isA360247 := proc(n)
local ifs, pidx, pe, meanAll, meanDist ;
if n = 1 then
return true ;
end if ;
ifs := ifactors(n)[2] ;
# list of prime indices with multiplicity
pidx := [] ;
for pe in ifs do
[numtheory[pi](op(1, pe)), op(2, pe)] ;
pidx := [op(pidx), %] ;
end do:
meanAll := add(op(1, pe)*op(2, pe), pe=pidx) / add(op(2, pe), pe=pidx) ;
meanDist := add(op(1, pe), pe=pidx) / nops(pidx) ;
if meanAll = meanDist then
true;
else
false;
end if;
end proc:
for n from 1 to 130 do
if isA360247(n) then
printf("%d, ", n) ;
end if;
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Mean[prix[#]]==Mean[Union[prix[#]]]&]
CROSSREFS
These partitions are counted by A360243. For median instead of mean the complement is A360248, counted by A360244. For greater instead of equal mean we have A360252, counted by A360250. A316413 = numbers whose prime indices have integer mean, distinct A326621.
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