A052335 -id:A052335 - 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

A276078 Numbers n in whose prime factorization no exponent of any prime(k) exceeds k.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98, 99, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121

Offset: 1

Antti Karttunen, Aug 18 2016

nonn

Numbers not divisible by p^(1+A000720(p)) for any prime p, where A000720(p) gives the index of prime p: 1 for 2, 2 for 3, 3 for 5, and so on.
Also Heinz numbers of integer partitions where the multiplicity of i does not exceed i for any i (A052335). Differs from A048103 in lacking {625, 1250, 1875, 3750, 4375, 5625, 6875, 8125, 8750, ...}. - Gus Wiseman, Mar 09 2019
Asymptotic density is Product_{i>=1} 1-prime(i)^(-1-i) = 0.72102334... - Amiram Eldar, Oct 20 2020

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positions of zeros in A276077.
Complement: A276079.
Sequence A276076 sorted into ascending order.
Cf. A000040, A000720.
Subsequence of A048103 from which it differs for the first time at n=451, where a(451) = 626, while A048103(451) = 625, a value missing from here.
Cf. A052335, A056239, A062457, A087153, A109298, A112798, A115584, A117144, A118914, A124010, A197987, A276078.
Cf. A324524, A324525, A324571, A324587, A324588.

Select[Range@ 121, Or[# == 1, AllTrue[FactorInteger[#], PrimePi[#1] >= #2 & @@ # &]] &] (* Michael De Vlieger, Jun 24 2017 *)

isok(n) = my(f=factor(n)); for (k=1, #f~, if (f[k, 2] > primepi(f[k, 1]), return(0))); return (1); \\ Michel Marcus, Jun 24 2017

is(n) = {my(t=1);forprime(p = 2, , t++; pp = p^t; if(n%pp==0, return(0)); if(pp > n, return(1)))} \\ David A. Corneth, Jun 24 2017

upto(n) = {my(v = vector(n,i,1), t=1, res=List()); forprime(p=2, , t++; pp = p^t; if(pp>n, break); for(i=1, n\pp, v[pp*i] = 0)); for(i=1, n, if(v[i]==1, listput(res, i))); res} \\ David A. Corneth, Jun 24 2017

from sympy import factorint, primepi
def ok(n):
    f = factorint(n)
    return all(f[i] <= primepi(i) for i in f)
print([n for n in range(1, 151) if ok(n)]) # Indranil Ghosh, Jun 24 2017

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}).

A115584 Number of partitions of n in which each part k occurs more than k times.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 4, 3, 6, 4, 7, 7, 8, 8, 12, 9, 15, 14, 17, 18, 24, 21, 29, 29, 35, 35, 46, 42, 56, 54, 65, 67, 81, 77, 98, 95, 115, 114, 139, 135, 164, 165, 190, 195, 230, 225, 272, 271, 313, 321, 370, 374, 433, 441, 501, 514, 589, 592, 681, 698, 778, 809, 907

Offset: 0

Vladeta Jovovic, Mar 09 2006

The Heinz numbers of these partitions are given by A325127. - Gus Wiseman, Apr 02 2019

			a(2) = 1 because we have [1,1]; a(10) = 4 because we have [2,2,2,2,2], [2,2,2,2,1,1], [2,2,2,1,1,1,1] and [1^10].
From _Gus Wiseman_, Apr 02 2019: (Start)
The initial terms count the following integer partitions:
   0: ()
   2: (11)
   3: (111)
   4: (1111)
   5: (11111)
   6: (222)
   6: (111111)
   7: (1111111)
   8: (2222)
   8: (22211)
   8: (11111111)
   9: (222111)
   9: (111111111)
  10: (22222)
  10: (222211)
  10: (2221111)
  10: (1111111111)
  11: (2222111)
  11: (22211111)
  11: (11111111111)
  12: (3333)
  12: (222222)
  12: (2222211)
  12: (22221111)
  12: (222111111)
  12: (111111111111)
(End)

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

Cf. A052335, A087153, A114639, A117144, A276429, A324572, A325127.

g:=product((1-x^k+x^(k*(k+1)))/(1-x^k),k=1..30): gser:=series(g,x=0,75): seq(coeff(gser,x,n),n=0..70); # Emeric Deutsch, Mar 12 2006
# 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+1..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Feb 09 2017

CoefficientList[ Series[ Product[(1 - x^k + x^(k(k + 1)))/(1 - x^k), {k, 14}], {x, 0, 66}], x] (* Robert G. Wilson v, Mar 12 2006 *)
Table[Length[Select[IntegerPartitions[n],And@@Table[Count[#,i]>i,{i,Union[#]}]&]],{n,0,30}] (* Gus Wiseman, Apr 02 2019 *)

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

More terms from Robert G. Wilson v and Emeric Deutsch, Mar 12 2006

A324588 Heinz numbers of integer partitions of n into perfect squares (A001156).

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 16, 23, 28, 32, 46, 49, 53, 56, 64, 92, 97, 98, 106, 112, 128, 151, 161, 184, 194, 196, 212, 224, 227, 256, 302, 311, 322, 343, 368, 371, 388, 392, 419, 424, 448, 454, 512, 529, 541, 604, 622, 644, 661, 679, 686, 736, 742, 776, 784, 827, 838

Offset: 1

Gus Wiseman, Mar 08 2019

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Also products of elements of A011757.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   4: {1,1}
   7: {4}
   8: {1,1,1}
  14: {1,4}
  16: {1,1,1,1}
  23: {9}
  28: {1,1,4}
  32: {1,1,1,1,1}
  46: {1,9}
  49: {4,4}
  53: {16}
  56: {1,1,1,4}
  64: {1,1,1,1,1,1}
  92: {1,1,9}
  97: {25}
  98: {1,4,4}

Cf. A001156, A011757, A033461, A052335, A056239, A062457, A078135, A112798, A117144, A118914, A276078.

Cf. A109298, A324524, A324525, A324571, A324572, A324587.

Select[Range[100],And@@Cases[FactorInteger[#],{p_,_}:>IntegerQ[Sqrt[PrimePi[p]]]]&]

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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A276078 Numbers n in whose prime factorization no exponent of any prime(k) exceeds k.

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

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