A324572 -id:A324572 - cp's OEIS frontend

A001156 Number of partitions of n into squares.

1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 8, 9, 10, 10, 12, 13, 14, 14, 16, 19, 20, 21, 23, 26, 27, 28, 31, 34, 37, 38, 43, 46, 49, 50, 55, 60, 63, 66, 71, 78, 81, 84, 90, 98, 104, 107, 116, 124, 132, 135, 144, 154, 163, 169, 178, 192, 201, 209, 220, 235, 247, 256

Offset: 0

N. J. A. Sloane

Number of partitions of n such that number of parts equal to k is multiple of k for all k. - Vladeta Jovovic, Aug 01 2004
Of course p_{4*square}(n)>0. In fact p_{4*square}(32n+28)=3 times p_{4*square}(8n+7) and p_{4*square}(72n+69) is even. These seem to be the only arithmetic properties the function p_{4*square(n)} possesses. Similar results hold for partitions into positive squares, distinct squares and distinct positive squares. - Michael David Hirschhorn, May 05 2005
The Heinz numbers of these partitions are given by A324588. - Gus Wiseman, Mar 09 2019

			p_{4*square}(23)=1 because 23 = 3^2 + 3^2 + 2^2 + 1^2 and there is no other partition of 23 into squares.
G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 2*x^7 +...
such that the g.f. A(x) satisfies the identity [_Paul D. Hanna_]:
A(x) = 1/((1-x)*(1-x^4)*(1-x^9)*(1-x^16)*(1-x^25)*...)
A(x) = 1 + x/(1-x) + x^4/((1-x)*(1-x^4)) + x^9/((1-x)*(1-x^4)*(1-x^9)) + x^16/((1-x)*(1-x^4)*(1-x^9)*(1-x^16)) + ...
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(14) = 6 integer partitions into squares are:
  (941)
  (911111)
  (44411)
  (44111111)
  (41111111111)
  (11111111111111)
while the a(14) = 6 integer partitions in which the multiplicity of k is a multiple of k for all k are:
  (333221)
  (33311111)
  (22222211)
  (2222111111)
  (221111111111)
  (11111111111111)
(End)

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).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301.
J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301. (Annotated scanned copy)
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.
M. D. Hirschhorn and J. A. Sellers, On a problem of Lehmer on partitions into squares, The Ramanujan Journal 8 (2004), 279-287.
F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sci. 16E, 237-240, 1997.
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.
Florentin Smarandache, Sequences of Numbers Involved in Unsolved Problems, arXiv:math/0604019 [math.GM], 2006.
Eric Weisstein's World of Mathematics, Partition
Eric Weisstein's World of Mathematics, Smarandache Sequences
Eric Weisstein's World of Mathematics, Square Number

Cf. A000041, A000161 (partitions into 2 squares), A000290, A033461, A131799, A218494, A285218, A304046.
Cf. A078134 (first differences).
Cf. A003108, A037444, A046042, A259792, A259793, A294529.
Row sums of A243148.
Euler trans. of A010052 (see also A308297).
Cf. A001462, A003114, A006141, A011757, A039900, A047993, A052335, A062457, A064174, A078135, A109298, A117144.
Cf. A324524, A324572, A324587, A324588.

a001156 = p (tail a000290_list) where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Oct 31 2012, Aug 14 2011

m:=70; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1/(1-x^(k^2)): k in [1..(m+2)]]) )); // G. C. Greubel, Nov 11 2018

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))))
    end:
a:= n-> b(n, isqrt(n)):
seq(a(n), n=0..120);  # Alois P. Heinz, May 30 2014

CoefficientList[ Series[Product[1/(1 - x^(m^2)), {m, 70}], {x, 0, 68}], x] (* Or *)
Join[{1}, Table[Length@PowersRepresentations[n, n, 2], {n, 68}]] (* Robert G. Wilson v, Apr 12 2005, revised Sep 27 2011 *)
f[n_] := Length@ IntegerPartitions[n, All, Range@ Sqrt@ n^2]; Array[f, 67] (* Robert G. Wilson v, Apr 14 2013 *)
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]]]]; a[n_] := b[n, Sqrt[n]//Floor]; Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Nov 02 2015, after Alois P. Heinz *)

{a(n)=polcoeff(1/prod(k=1, sqrtint(n+1), 1-x^(k^2)+x*O(x^n)), n)} \\ Paul D. Hanna, Mar 09 2012

{a(n)=polcoeff(1+sum(m=1, sqrtint(n+1), x^(m^2)/prod(k=1, m, 1-x^(k^2)+x*O(x^n))), n)} \\ Paul D. Hanna, Mar 09 2012

G.f.: Product_{m>=1} 1/(1-x^(m^2)).
G.f.: Sum_{n>=0} x^(n^2) / Product_{k=1..n} (1 - x^(k^2)). - Paul D. Hanna, Mar 09 2012
a(n) = (1/n)*Sum_{k=1..n} A035316(k)*a(n-k). - Vladeta Jovovic, Nov 20 2002
a(n) = f(n,1,3) with f(x,y,z) = if xReinhard Zumkeller, Nov 08 2009
Conjecture (Jan Bohman, Carl-Erik Fröberg, Hans Riesel, 1979): a(n) ~ c * n^(-alfa) * exp(beta*n^(1/3)), where c = 1/18.79656, beta = 3.30716, alfa = 1.16022. - Vaclav Kotesovec, Aug 19 2015
From Vaclav Kotesovec, Dec 29 2016: (Start)
Correct values of these constants are:
1/c = sqrt(3) * (4*Pi)^(7/6) / Zeta(3/2)^(2/3) = 17.49638865935104978665...
alfa = 7/6 = 1.16666666666666666...
beta = 3/2 * (Pi/2)^(1/3) * Zeta(3/2)^(2/3) = 3.307411783596651987...
a(n) ~ 3^(-1/2) * (4*Pi*n)^(-7/6) * Zeta(3/2)^(2/3) * exp(2^(-4/3) * 3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3)). [Hardy & Ramanujan, 1917]
(End)

More terms from Eric W. Weisstein

More terms from Gh. Niculescu (ghniculescu(AT)yahoo.com), Oct 08 2006

A033461 Number of partitions of n into distinct squares.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

"WEIGH" transform of squares A000290.
a(n) = 0 for n in {A001422}, a(n) > 0 for n in {A003995}. - Alois P. Heinz, May 14 2014
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
From Gus Wiseman, Mar 09 2019: (Start)
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)
(22)
(221)
(333)
(3331)
(33322)
(333221)
(4444)
(44441)
(444422)
(4444221)
(55555)
(4444333)
(555551)
(44443331)
(5555522)
(444433322)
(55555221)
(4444333221)
The case where the distinct parts are taken in decreasing order is A324572, with Heinz numbers given by A324571.
(End)

			a(50)=3 because we have [1,4,9,36], [1,49], and [9,16,25]. - _Emeric Deutsch_, Jan 26 2016
From _Gus Wiseman_, Mar 09 2019: (Start)
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)

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 288-289.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
M. Brack, M. V. N. Murthy, and J. Bartel, Application of semiclassical methods to number theory, University of Regensburg (Germany, 2020).
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Vaclav Kotesovec, Graph - The asymptotic ratio.
M. V. N. Murthy, Matthias Brack, Rajat K. Bhaduri, and Johann Bartel, Semi-classical analysis of distinct square partitions, arXiv:1808.05146 [cond-mat.stat-mech], 2018.

Cf. A001422, A003995, A078434, A242434 (the same for compositions), A279329.
Cf. A001156 (non-strict case), A001462, A005117, A052335, A078135, A109298, A114638, A117144, A324571, A324572, A324587, A324588.
Row sums of A341040.

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)):
seq(a(n), n=0..100);  # Alois P. Heinz, May 14 2014

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 *)

a(n)=polcoeff(prod(k=1,sqrt(n),1+x^k^2), n)

first(n)=Vec(prod(k=1,sqrtint(n),1+'x^k^2,O('x^(n+1))+1)) \\ Charles R Greathouse IV, Sep 03 2015

from functools import cache
from sympy.core.power import isqrt
@cache
def b(n,i):
  # Code after Alois P. Heinz
  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

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

More terms from Michael Somos

A064174 Number of partitions of n with nonnegative rank.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 12, 17, 23, 31, 42, 56, 73, 96, 125, 161, 207, 265, 336, 426, 536, 672, 840, 1046, 1296, 1603, 1975, 2425, 2970, 3628, 4417, 5367, 6503, 7861, 9482, 11412, 13702, 16423, 19642, 23447, 27938, 33231, 39453, 46767, 55342, 65386, 77135

Offset: 1

Vladeta Jovovic, Sep 20 2001

nonn

The rank of a partition is the largest summand minus the number of summands.
This sequence (up to proof) equals "partitions of 2n with even number of parts, ending in 1, with max descent of 1, where the number of odd parts in odd places equals the number of odd parts in even places. (See link and 2nd Mathematica line.) - Wouter Meeussen, Mar 29 2013
Number of partitions p of n such that max(max(p), number of parts of p) is a part of p. - Clark Kimberling, Feb 28 2014
From Gus Wiseman, Mar 09 2019: (Start)
Also the number of integer partitions of n with maximum part greater than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324521. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)
            (21)  (22)  (32)   (33)   (43)    (44)
                  (31)  (41)   (42)   (52)    (53)
                        (311)  (51)   (61)    (62)
                               (321)  (322)   (71)
                               (411)  (331)   (332)
                                      (421)   (422)
                                      (511)   (431)
                                      (4111)  (521)
                                              (611)
                                              (4211)
                                              (5111)
Also the number of integer partitions of n with maximum part less than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324562. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (1)  (11)  (21)   (22)    (221)    (222)     (322)      (332)
             (111)  (211)   (311)    (321)     (331)      (2222)
                    (1111)  (2111)   (2211)    (2221)     (3221)
                            (11111)  (3111)    (3211)     (3311)
                                     (21111)   (4111)     (4211)
                                     (111111)  (22111)    (22211)
                                               (31111)    (32111)
                                               (211111)   (41111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

			a(20) = p(19) - p(15) + p(8) = 490 - 176 + 22 = 336.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
Cristina Ballantine and Mircea Merca, Bisected theta series, least r-gaps in partitions, and polygonal numbers, arXiv:1710.05960 [math.CO], 2017.
Rekha Biswal, bijection between number of partitions of 2n satisfying certain conditions with number of partitions of n, Mathoverflow.
Mircea Merca, Rank partition functions and truncated theta identities, arXiv:2006.07705 [math.CO], 2020.

Cf. A063995, A064173 (complement).
Row sums of triangle A105806.
Cf. A003114, A006141, A039900, A047993, A090858, A106529, A133121.
Cf. A324516, A324518, A324520, A324521, A324522, A324560, A324562, A324572.

f:= n -> add((-1)^(k+1)*combinat:-numbpart(n-(3*k^2-k)/2),k=1..floor((1+sqrt(24*n+1))/6)):
map(f, [$1..100]); # Robert Israel, Aug 03 2015

Table[Count[IntegerPartitions[n], q_ /; First[q] >= Length[q]], {n, 16}]
(* also *)
Table[Count[IntegerPartitions[2n],q_/;Last[q]===1 && Max[q-PadRight[Rest[q],Length[q]]]<=1 && Count[First/@Partition[q,2],?OddQ]==Count[Last/@Partition[q,2],?OddQ]],{n,16}]
(* also *)
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[Max[p], Length[p]]]], {n, 50}] (* Clark Kimberling, Feb 28 2014 *)

{a(n) = my(A=1); A = sum(m=0,n,x^m*prod(k=1,m,(1-x^(m+k-1))/(1-x^k +x*O(x^n)))); polcoeff(A,n)}
for(n=1,60,print1(a(n),", ")) \\ Paul D. Hanna, Aug 03 2015

my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2))) \\ Seiichi Manyama, May 21 2023

a(n) = (A000041(n) + A047993(n))/2.
a(n) = p(n-1) - p(n-5) + p(n-12) - ... -(-1)^k*p(n-(3*k^2-k)/2) + ..., where p() is A000041(). - Vladeta Jovovic, Aug 04 2004
G.f.: Sum_{n>=1} x^n * Product_{k=1..n} (1 - x^(n+k-1))/(1 - x^k). - Paul D. Hanna, Aug 03 2015
A064173(n) + a(n) = A000041(n). - R. J. Mathar, Feb 22 2023
G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2). - Seiichi Manyama, May 21 2023

Mathematica programs modified by Clark Kimberling, Feb 12 2014

A109298 Primal codes of finite idempotent functions on positive integers.

Original entry on oeis.org

1, 2, 9, 18, 125, 250, 1125, 2250, 2401, 4802, 21609, 43218, 161051, 300125, 322102, 600250, 1449459, 2701125, 2898918, 4826809, 5402250, 9653618, 20131375, 40262750, 43441281, 86882562, 181182375, 362364750, 386683451, 410338673, 603351125, 773366902, 820677346

Offset: 1

Jon Awbrey, Jul 06 2005

nonn

Finite idempotent functions are identity maps on finite subsets, counting the empty function as the idempotent on the empty set.
From Gus Wiseman, Mar 09 2019: (Start)
Also numbers whose ordered prime signature is equal to the distinct prime indices in increasing order. A prime index of n is a number m such that prime(m) divides n. The ordered prime signature (A124010) is the sequence of multiplicities (or exponents) in a number's prime factorization, taken in order of the prime base. The case where the prime indices are taken in decreasing order is A324571.
Also numbers divisible by prime(k) exactly k times for each prime index k. These are a kind of self-describing numbers (cf. A001462, A304679).
Also Heinz numbers of integer partitions where the multiplicity of m is m for all m in the support (counted by A033461). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also products of distinct elements of A062457. For example, 43218 = prime(1)^1 * prime(2)^2 * prime(4)^4.
(End)

			Writing (prime(i))^j as i:j, we have the following table of examples:
Primal Codes of Finite Idempotent Functions on Positive Integers
` ` ` 1 = { }
` ` ` 2 = 1:1
` ` ` 9 = ` ` 2:2
` ` `18 = 1:1 2:2
` ` 125 = ` ` ` ` 3:3
` ` 250 = 1:1 ` ` 3:3
` `1125 = ` ` 2:2 3:3
` `2250 = 1:1 2:2 3:3
` `2401 = ` ` ` ` ` ` 4:4
` `4802 = 1:1 ` ` ` ` 4:4
` 21609 = ` ` 2:2 ` ` 4:4
` 43218 = 1:1 2:2 ` ` 4:4
`161051 = ` ` ` ` ` ` ` ` 5:5
`300125 = ` ` ` ` 3:3 4:4
`322102 = 1:1 ` ` ` ` ` ` 5:5
`600250 = 1:1 ` ` 3:3 4:4
From _Gus Wiseman_, Mar 09 2019: (Start)
The sequence of terms together with their prime indices begins as follows. For example, we have 18: {1,2,2} because 18 = prime(1) * prime(2) * prime(2) has prime signature {1,2} and the distinct prime indices are also {1,2}.
       1: {}
       2: {1}
       9: {2,2}
      18: {1,2,2}
     125: {3,3,3}
     250: {1,3,3,3}
    1125: {2,2,3,3,3}
    2250: {1,2,2,3,3,3}
    2401: {4,4,4,4}
    4802: {1,4,4,4,4}
   21609: {2,2,4,4,4,4}
   43218: {1,2,2,4,4,4,4}
  161051: {5,5,5,5,5}
  300125: {3,3,3,4,4,4,4}
  322102: {1,5,5,5,5,5}
  600250: {1,3,3,3,4,4,4,4}
(End)

David A. Corneth, Table of n, a(n) for n = 1..10000
J. Awbrey, Riffs and Rotes

Cf. A076954, A106177, A108352, A108371, A109297, A109301.
Cf. A001156, A033461, A056239, A062457, A112798, A118914, A124010 (ordered prime signature), A181819, A276078, A304679.
Cf. A324524, A324525, A324570, A324571, A324572, A324587, A324588.
Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

Select[Range[10000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]==k]&]

is(n) = my(f = factor(n)); for(i = 1, #f~, if(prime(f[i, 2]) != f[i, 1], return(0))); 1 \\ David A. Corneth, Mar 09 2019

Sum_{n>=1} 1/a(n) = Product_{n>=1} (1 + 1/prime(n)^n) = 1.6807104966... - Amiram Eldar, Jan 03 2021

Offset set to 1, missing terms inserted and more terms added by Alois P. Heinz, Mar 08 2019

A114640 Number of partitions of n such that the set of parts and the set of multiplicities of parts are equal.

Original entry on oeis.org

1, 1, 0, 0, 2, 1, 1, 0, 1, 1, 3, 2, 3, 3, 5, 0, 4, 5, 2, 3, 8, 6, 5, 10, 9, 9, 16, 14, 12, 16, 17, 10, 17, 15, 16, 19, 35, 17, 34, 37, 40, 31, 54, 36, 60, 61, 58, 63, 88, 58, 88, 87, 91, 84, 115, 93, 116, 108, 115, 130, 190, 143, 165, 214, 219, 200, 255, 240

Offset: 0

Vladeta Jovovic, Feb 18 2006

nonn

The Heinz numbers of these partitions are given by A109297. - Gus Wiseman, Apr 02 2019

			From _Gus Wiseman_, Apr 02 2019: (Start)
The initial terms count the following integer partitions:
   0: ()
   1: (1)
   4: (22)
   4: (211)
   5: (221)
   6: (3111)
   8: (41111)
   9: (333)
  10: (511111)
  10: (3331)
  10: (322111)
  11: (332111)
  11: (322211)
  12: (6111111)
  12: (4221111)
  12: (33222)
  13: (33322)
  13: (333211)
  13: (332221)
  14: (71111111)
  14: (52211111)
  14: (4421111)
  14: (4222211)
  14: (333221)
(End)

Cf. A052335, A087153, A109297, A114639, A115584, A117144, A276429, A324572, A325132, A336031.

Table[Length[Select[IntegerPartitions[n],Union[#]==Union[Length/@Split[#]]&]],{n,0,30}] (* Gus Wiseman, Apr 02 2019 *)

More terms from Alois P. Heinz, Aug 09 2016

A087153 Number of partitions of n into nonsquares.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 5, 5, 8, 9, 13, 15, 20, 24, 30, 37, 47, 55, 71, 83, 103, 123, 151, 178, 218, 257, 310, 366, 440, 515, 617, 722, 857, 1003, 1184, 1380, 1625, 1889, 2214, 2570, 3000, 3472, 4042, 4669, 5414, 6244, 7221, 8303, 9583, 10998, 12655, 14502

Offset: 0

Reinhard Zumkeller, Aug 21 2003

nonn

Also, number of partitions of n where there are fewer than k parts equal to k for all k. - Jon Perry and Vladeta Jovovic, Aug 04 2004. E.g. a(8)=5 because we have 8=6+2=5+3=4+4=3+3+2.
Convolution of A276516 and A000041. - Vaclav Kotesovec, Dec 30 2016
From Gus Wiseman, Apr 02 2019: (Start)
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The Heinz numbers of the integer partitions described in Perry and Jovovic's comment are given by A325128, while the Heinz numbers of the integer partitions described in the name are given by A325129. In the former case, the first 10 terms count the following integer partitions:
  ()  (2)  (3)  (4)  (5)   (6)   (7)   (8)    (9)
                     (32)  (33)  (43)  (44)   (54)
                           (42)  (52)  (53)   (63)
                                       (62)   (72)
                                       (332)  (432)
while in the latter case they count the following:
  ()  (2)  (3)  (22)  (5)   (6)    (7)    (8)     (63)
                      (32)  (33)   (52)   (53)    (72)
                            (222)  (322)  (62)    (333)
                                          (332)   (522)
                                          (2222)  (3222)
(End)

			n=7: 2+5 = 2+2+3 = 7: a(7)=3;
n=8: 2+6 = 2+2+2+2 = 2+3+3 = 3+5 = 8: a(8)=5;
n=9: 2+7 = 2+2+5 = 2+2+2+3 = 3+3+3 = 3+6: a(9)=5.

G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. See page 48.

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Daniel I. A. Cohen, PIE-sums: a combinatorial tool for partition theory. J. Combin. Theory Ser. A 31 (1981), no. 3, 223--236. MR0635367 (82m:10026). See Cor. 5. - _N. J. A. Sloane_, Mar 27 2012
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.

Cf. A087154, A001156, A000009, A000037, A052335 (<=k parts of k).
Cf. A115584, A172151, A225044, A264393, A276516.
Cf. A033461, A114639, A117144, A276429, A324572, A324588, A325128, A325129.

a087153 = p a000037_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Apr 25 2013

g:=product((1-x^(i^2))/(1-x^i),i=1..70):gser:=series(g,x=0,60):seq(coeff(gser,x^n),n=1..53); # Emeric Deutsch, Feb 09 2006

nn=54; CoefficientList[ Series[ Product[ Sum[x^(i*j), {j, 0, i - 1}], {i, 1, nn}], {x, 0, nn}], x] (* Robert G. Wilson v, Aug 05 2004 *)
nmax = 100; CoefficientList[Series[Product[(1 - x^(k^2))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 29 2016 *)

first(n)=my(x='x+O('x^(n+1))); Vec(prod(m=1,sqrtint(n), (1-x^m^2)/(1-x^m))*prod(m=sqrtint(n)+1,n,1/(1-x^m))) \\ Charles R Greathouse IV, Aug 28 2016

G.f.: Product_{m>0} (1-x^(m^2))/(1-x^m). - Vladeta Jovovic, Aug 21 2003
a(n) = (1/n)*Sum_{k=1..n} (A000203(k)-A035316(k))*a(n-k), a(0)=1. - Vladeta Jovovic, Aug 21 2003
G.f.: Product_{i>=1} (Sum_{j=0..i-1} x^(i*j)). - Jon Perry, Jul 26 2004
a(n) ~ exp(Pi*sqrt(2*n/3) - 3^(1/4) * Zeta(3/2) * n^(1/4) / 2^(3/4) - 3*Zeta(3/2)^2/(32*Pi)) * sqrt(Pi) / (2^(3/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Dec 30 2016

Zeroth term added by Franklin T. Adams-Watters, Jan 25 2010

A114639 Number of partitions of n such that the set of parts and the set of multiplicities of parts are disjoint.

Original entry on oeis.org

1, 0, 2, 2, 2, 3, 5, 4, 7, 7, 13, 16, 19, 23, 33, 34, 44, 58, 63, 80, 101, 112, 139, 171, 196, 234, 288, 328, 394, 478, 545, 658, 777, 881, 1050, 1236, 1414, 1666, 1936, 2216, 2592, 3018, 3428, 3992, 4604, 5243, 6069, 6986, 7951, 9139, 10447, 11892, 13625

Offset: 0

Vladeta Jovovic, Feb 18 2006

nonn

The Heinz numbers of these partitions are given by A325131. - Gus Wiseman, Apr 02 2019

			From _Gus Wiseman_, Apr 02 2019: (Start)
The a(2) = 2 through a(9) = 7 partitions:
  (2)   (3)    (4)     (5)      (6)       (7)        (8)         (9)
  (11)  (111)  (1111)  (32)     (33)      (43)       (44)        (54)
                       (11111)  (42)      (52)       (53)        (63)
                                (222)     (1111111)  (62)        (72)
                                (111111)             (2222)      (432)
                                                     (3311)      (222111)
                                                     (11111111)  (111111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A052335, A087153, A114640, A115584, A117144, A276429, A324572, A325130, A325131, A336032.

b:= proc(n, i, p, m) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1, p, select(x-> x x<=n-i*j, p union {i}),
         select(x-> x b(n$2, {}$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 09 2016

b[n_, i_, p_, m_] := b[n, i, p, m] = If[n == 0, 1, If[i<1, 0, b[n, i-1, p, Select[m, #Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Length/@Split[#]]=={}&]],{n,0,30}] (* Gus Wiseman, Apr 02 2019 *)

a(0)=1 prepended and more terms from Alois P. Heinz, Aug 09 2016

A117144 Partitions of n in which each part k occurs at least k times.

Original entry on oeis.org

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

Offset: 0

Emeric Deutsch, Mar 06 2006

nonn

The Heinz numbers of these integer partitions are given by A324525. - Gus Wiseman, Mar 09 2019

			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].
From _Gus Wiseman_, Mar 09 2019: (Start)
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)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A001462, A003114, A006141, A033461, A039900, A047993, A052335, A064174, A090858, A114638, A115584, A276078, A280204.

Cf. A324518, A324520, A324525, A324572.

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):
seq(a(n), n=0..80);  # Alois P. Heinz, Dec 28 2016

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 *)

G.f.: Product_{k>=1} (1-x^k+x^(k^2))/(1-x^k).

A276429 Number of partitions of n containing no part i of multiplicity i.

Original entry on oeis.org

1, 0, 2, 2, 3, 5, 8, 9, 16, 19, 29, 36, 53, 65, 92, 115, 154, 195, 257, 318, 419, 516, 663, 821, 1039, 1277, 1606, 1963, 2441, 2978, 3675, 4454, 5469, 6603, 8043, 9688, 11732, 14066, 16963, 20260, 24310, 28953, 34586, 41047, 48857, 57802, 68528, 80862, 95534, 112388, 132391

Offset: 0

Emeric Deutsch, Sep 19 2016

nonn

The Heinz numbers of these partitions are given by A325130. - Gus Wiseman, Apr 02 2019

			a(4) = 3 because we have [1,1,1,1], [1,1,2], and [4]; the partitions [1,3], [2,2] do not qualify.
From _Gus Wiseman_, Apr 02 2019: (Start)
The a(2) = 2 through a(7) = 9 partitions:
  (2)   (3)    (4)     (5)      (6)       (7)
  (11)  (111)  (211)   (32)     (33)      (43)
               (1111)  (311)    (42)      (52)
                       (2111)   (222)     (511)
                       (11111)  (411)     (3211)
                                (3111)    (4111)
                                (21111)   (31111)
                                (111111)  (211111)
                                          (1111111)
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..12782 (terms 0..5000 from Alois P. Heinz)

Cf. A052335, A087153, A114639, A115584, A117144, A276427, A324572, A325130, A325131, A336269.

g := product(1/(1-x^i)-x^(i^2), i = 1 .. 100): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(i=j, 0, b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 19 2016

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[If[i == j, x, 1]*b[n - i*j, i - 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n][[1]], {n, 0, 60}] (* Jean-François Alcover, Nov 28 2016 after Alois P. Heinz's Maple code for A276427 *)
Table[Length[Select[IntegerPartitions[n],And@@Table[Count[#,i]!=i,{i,Union[#]}]&]],{n,0,30}] (* Gus Wiseman, Apr 02 2019 *)

a(n) = A276427(n,0).

G.f.: g(x) = Product_{i>=1} (1/(1-x^i) - x^{i^2}).

A324525 Numbers divisible by prime(k)^k for each prime index k.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 27, 32, 36, 54, 64, 72, 81, 108, 125, 128, 144, 162, 216, 243, 250, 256, 288, 324, 432, 486, 500, 512, 576, 625, 648, 729, 864, 972, 1000, 1024, 1125, 1152, 1250, 1296, 1458, 1728, 1944, 2000, 2048, 2187, 2250, 2304, 2401, 2500, 2592

Offset: 1

Gus Wiseman, Mar 08 2019

nonn

These are a kind of self-describing numbers (cf. A001462, A304679).
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. The prime signature of a number is the multiset of multiplicities (or exponents) in its prime factorization.
Also Heinz numbers of integer partitions where the multiplicity of k is at least k (A117144). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins as follows. For example, 36 = prime(1) * prime(1) * prime(2) * prime(2) is a term because the prime multiplicities are {2,2}, which are greater than or equal to the prime indices {1,2}.
    1: {}
    2: {1}
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   16: {1,1,1,1}
   18: {1,2,2}
   27: {2,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   54: {1,2,2,2}
   64: {1,1,1,1,1,1}
   72: {1,1,1,2,2}
   81: {2,2,2,2}
  108: {1,1,2,2,2}
  125: {3,3,3}
  128: {1,1,1,1,1,1,1}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001156, A033461, A056239, A062457, A112798, A117144, A118914 (prime signature), A124010, A181819, A276078, A304679.
Cf. A109298, A324524, A324571, A324572, A324587, A324588.
Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

q:= n-> andmap(i-> i[2]>=numtheory[pi](i[1]), ifactors(n)[2]):
select(q, [$1..3000])[];  # Alois P. Heinz, Mar 08 2019

Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k>=PrimePi[p]]&]
seq[max_] := Module[{ps = {2}, p, s = {1}, s1, s2, emax}, While[ps[[-1]]^Length[ps] < max, AppendTo[ps, NextPrime[ps[[-1]]]]]; Do[p = ps[[k]]; emax = Floor[Log[p, max]]; s1 = Join[{1}, p^Range[k, emax]]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= max &]; s = Union[s, s2], {k, 1, Length[ps]}]; s]; seq[3000] (* Amiram Eldar, Nov 23 2020 *)

Closed under multiplication.

Sum_{n>=1} 1/a(n) = Product_{k>=1} 1 + 1/(prime(k)^(k-1) * (prime(k)-1)) = 2.35782843100111139159... - Amiram Eldar, Nov 23 2020

cp's OEIS Frontend

A001156 Number of partitions of n into squares.

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A033461 Number of partitions of n into distinct squares.

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A064174 Number of partitions of n with nonnegative rank.

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A109298 Primal codes of finite idempotent functions on positive integers.

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A114640 Number of partitions of n such that the set of parts and the set of multiplicities of parts are equal.

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A087153 Number of partitions of n into nonsquares.

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