A001156 -id:A001156 - cp's OEIS frontend

A131799 Number of partitions of n into parts that are squares or cubes.

1, 1, 1, 1, 2, 2, 2, 2, 4, 5, 5, 5, 7, 8, 8, 8, 12, 14, 15, 15, 19, 21, 22, 22, 28, 33, 35, 37, 43, 48, 50, 52, 61, 69, 74, 78, 90, 98, 103, 107, 122, 135, 143, 152, 170, 186, 194, 203, 225, 247, 261, 275, 305, 330, 348, 362, 396, 429, 454, 477, 519, 561, 590, 618, 666, 717

Offset: 0

Reinhard Zumkeller, Jul 16 2007

nonn

a(n) = A078635(n) for n < 32 = 2^5.

			a(10) = #{9+1, 8+1+1, 4+4+1+1, 4+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1} = 5.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000041, A001156, A002760, A003108, A078635.

nmax = 65; c2max = nmax^(1/2); c3max = nmax^(1/3);
s = Flatten[{Table[n^2, {n, 1, c2max}]}~Join~{Table[n^3, {n, 1, c3max}]}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Jul 31 2020 *)

G.f.: Product_{k>=1} (1 - x^(k^6)) / ((1 - x^(k^2)) * (1 - x^(k^3))). - Vaclav Kotesovec, Jan 12 2017

a(0)=1 prepended by Ilya Gutkovskiy, Jan 11 2017

A280129 Expansion of Product_{k>=2} (1 + x^(k^2)).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 2, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 1, 3, 0, 0, 1, 1, 1, 0, 0, 1, 3, 0, 0, 2, 2, 0, 1, 2, 0, 1, 1, 2, 1

Offset: 0

Ilya Gutkovskiy, Dec 26 2016

nonn

Number of partitions of n into distinct squares > 1.

			G.f. = 1 + x^4 + x^9 + x^13 + x^16 + x^20 + 2*x^25 + 2*x^29 + x^34 + x^36 + ...
a(25) = 2 because we have [25] and [16, 9].

Cf. A001156, A033461, A078134.

nmax = 115; CoefficientList[Series[Product[1 + x^k^2, {k, 2, nmax}], {x, 0, nmax}], x]

{a(n) = if(n < 0, 0, polcoeff( prod(k=2, sqrtint(n), 1 + x^k^2 + x*O(x^n)), n))}; /* Michael Somos, Dec 26 2016 */

G.f.: Product_{k>=2} (1 + x^(k^2)).
From Vaclav Kotesovec, Dec 26 2016: (Start)
a(n) = Sum_{k=0..n} (-1)^(n-k) * A033461(k).
a(n) + a(n-1) = A033461(n).
a(n) ~ A033461(n)/2.
(End)

A281541 Expansion of Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) / Product_{j>=1} (1 - x^(j^2)).

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 15, 19, 23, 27, 34, 41, 47, 53, 64, 75, 86, 96, 113, 129, 145, 159, 182, 206, 229, 252, 284, 318, 349, 380, 423, 468, 513, 555, 616, 676, 736, 791, 869, 949, 1026, 1103, 1202, 1310, 1408, 1506, 1631, 1766, 1896, 2020, 2185, 2354, 2525, 2680, 2882, 3094, 3305, 3506, 3751, 4023, 4281, 4537

Offset: 1

Ilya Gutkovskiy, Jan 23 2017

nonn

Total number of parts in all partitions of n into squares.

Convolution of A001156 and A046951.

			a(8) = 15 because we have [4, 4], [4, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1] and 2 + 5 + 8 = 15.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index entries for related partition-counting sequences

Cf. A000290, A001156, A006128, A046951, A243148.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n], (s->
      `if`(s>n, 0, (p->p+[0, p[1]])(b(n-s, i))))(i^2)+b(n, i-1))
    end:
a:= n-> b(n, isqrt(n))[2]:
seq(a(n), n=1..70);  # Alois P. Heinz, Sep 19 2018

nmax = 63; Rest[CoefficientList[Series[Sum[x^i^2/(1 - x^i^2), {i, 1, nmax}]/Product[1 - x^j^2, {j, 1, nmax}], {x, 0, nmax}], x]]

G.f.: Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) / Product_{j>=1} (1 - x^(j^2)).
a(n) = Sum_{k=1..n} k * A243148(n,k). - Alois P. Heinz, Sep 19 2018
a(n) ~ exp(3 * 2^(-4/3) * zeta(3/2)^(2/3) * (Pi*n)^(1/3)) * sqrt(Pi/3) / (12*sqrt(n)). - Vaclav Kotesovec, Sep 15 2021

A284345 Number of partitions of n into squares dividing n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 6, 1, 3, 1, 6, 1, 1, 1, 7, 2, 1, 4, 8, 1, 1, 1, 15, 1, 1, 1, 27, 1, 1, 1, 11, 1, 1, 1, 12, 6, 1, 1, 28, 2, 3, 1, 14, 1, 7, 1, 15, 1, 1, 1, 16, 1, 1, 8, 46, 1, 1, 1, 18, 1, 1, 1, 114, 1, 1, 4, 20, 1, 1, 1, 66, 11, 1, 1, 22, 1, 1, 1, 23, 1, 11, 1, 24, 1, 1, 1, 91, 1, 3, 12, 67

Offset: 0

Ilya Gutkovskiy, Mar 25 2017

nonn

			a(8) = 3 because 8 has 4 divisors {1, 2, 4, 8} among which 2 are squares {1, 4} therefore we have [4, 4], [4, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

Cf. A000290, A001156, A018818, A046951, A066882, A161148, A225244, A284289.

with(numtheory):
a:= proc(n) option remember; local b, l; l, b:=
      sort(select(issqr, [divisors(n)[]])),
      proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
        b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
      end; b(n, nops(l))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 30 2017

Join[{1}, Table[d = Divisors[n]; Coefficient[Series[Product[1/(1 - Boole[Mod[DivisorSigma[0, d[[k]]], 2] == 1] x^d[[k]]), {k, Length[d]}], {x, 0, n}], x, n], {n, 1, 100}]]

a(n) = [x^n] Product_{d^2|n} 1/(1 - x^(d^2)).
a(n) = 1 if n is a squarefree.
a(n) = 2 if n is a square of prime.

A285047 Expansion of Product_{k>=1} 1/(1-x^(k^2))^k.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 3, 3, 6, 9, 9, 9, 13, 19, 19, 19, 28, 37, 43, 43, 57, 69, 81, 81, 100, 132, 150, 160, 184, 236, 260, 280, 319, 391, 460, 490, 565, 657, 771, 811, 922, 1084, 1243, 1363, 1510, 1781, 1985, 2185, 2388, 2775, 3159, 3439, 3832, 4335, 4963, 5323

Offset: 0

Vaclav Kotesovec, Apr 15 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000219, A001156, A023871, A266941, A281790.

nmax=100; CoefficientList[Series[Product[1/(1-x^(k^2))^k, {k,1,nmax}], {x,0,nmax}], x]

log(a(n)) ~ Pi*sqrt(n/3).

A324587 Heinz numbers of integer partitions of n into distinct perfect squares (A033461).

Original entry on oeis.org

1, 2, 7, 14, 23, 46, 53, 97, 106, 151, 161, 194, 227, 302, 311, 322, 371, 419, 454, 541, 622, 661, 679, 742, 827, 838, 1009, 1057, 1082, 1193, 1219, 1322, 1358, 1427, 1589, 1619, 1654, 1879, 2018, 2114, 2143, 2177, 2231, 2386, 2437, 2438, 2741, 2854, 2933

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 distinct elements of A011757.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    7: {4}
   14: {1,4}
   23: {9}
   46: {1,9}
   53: {16}
   97: {25}
  106: {1,16}
  151: {36}
  161: {4,9}
  194: {1,25}
  227: {49}
  302: {1,36}
  311: {64}
  322: {1,4,9}
  371: {4,16}
  419: {81}
  454: {1,49}
  541: {100}

Cf. A001156, A005117, A011757, A033461, A052335, A056239, A062457, A078135, A112798, A117144, A276078.

Cf. A109298, A324524, A324525, A324571, A324572, A324588.

Select[Range[1000],And@@Cases[FactorInteger[#],{p_,k_}:>k==1&&IntegerQ[Sqrt[PrimePi[p]]]]&]

A351982 Number of integer partitions of n into prime parts with prime multiplicities.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 3, 0, 1, 1, 3, 3, 3, 0, 1, 4, 5, 5, 3, 3, 5, 8, 5, 5, 6, 8, 8, 11, 7, 8, 10, 17, 14, 14, 12, 17, 17, 21, 18, 23, 20, 28, 27, 31, 27, 36, 32, 35, 37, 46, 41, 52, 45, 60, 58, 63, 59, 78, 71, 76, 81, 87, 80, 103, 107, 113, 114, 127

Offset: 0

Gus Wiseman, Mar 18 2022

nonn

			The partitions for n = 4, 6, 10, 19, 20, 25:
  (22)  (33)   (55)     (55333)     (7733)       (55555)
        (222)  (3322)   (55522)     (77222)      (77722)
               (22222)  (3333322)   (553322)     (5533333)
                        (33322222)  (5522222)    (5553322)
                                    (332222222)  (55333222)
                                                 (55522222)
                                                 (333333322)
                                                 (3333322222)

The version for just prime parts is A000607, ranked by A076610.
The version for just prime multiplicities is A055923, ranked by A056166.
For odd instead of prime we have A117958, ranked by A352142.
The constant case is A230595, ranked by A352519.
Allowing any multiplicity > 1 gives A339218, ranked by A352492.
These partitions are ranked by A346068.
The non-constant case is A352493, ranked by A352518.
A000040 lists the primes.
A001221 counts constant partitions of prime length, ranked by A053810.
A001694 lists powerful numbers, counted A007690, weak A052485.
A038499 counts partitions of prime length.
A101436 counts parts of prime signature that are themselves prime.
A112798 lists prime indices, reverse A296150, sum A056239.
A124010 gives prime signature, sorted A118914, sum A001222.
A257994 counts prime indices that are prime, nonprime A330944.
Cf. A000720, A000961, A001156, A011757, A052335, A164336, A320628.

Table[Length[Select[IntegerPartitions[n], And@@PrimeQ/@#&&And@@PrimeQ/@Length/@Split[#]&]],{n,0,30}]

A092362 Number of partitions of n^2 into squares greater than 1.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 8, 11, 28, 44, 94, 167, 354, 643, 1314, 2412, 4792, 8981, 17374, 32566, 62008, 115702, 217040, 402396, 745795, 1372266, 2517983, 4595652, 8354350, 15125316, 27265107, 48972467, 87584837, 156119631, 277152178, 490437445, 864534950

Offset: 0

Reinhard Zumkeller, Mar 19 2004

nonn

a(n) = A078134(A000290(n)).

			a(6) = 5: 6^2 = 36 = 16+16+4 = 16+4+4+4+4+4 = 9+9+9+9 = 4+4+4+4+4+4+4+4+4.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A001156, A037444.

Cf. A093115, A093116.

b:=proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<2, 0, b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i))))
   end:
a:= n-> b(n^2, n):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 15 2013

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<2, 0, b[n, i-1] + If[i^2>n, 0, b[n-i^2, i]]]]; a[n_] := b[n^2, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

a(n) ~ exp(3*Pi^(1/3) * Zeta(3/2)^(2/3) * n^(2/3) / 2^(4/3)) * Zeta(3/2)^(4/3) / (2^(11/3) * sqrt(3) * Pi^(5/6) * n^(11/3)). - Vaclav Kotesovec, Apr 10 2017

Corrected a(0) and more terms from Alois P. Heinz, Apr 15 2013

A111178 Number of partitions of n into positive numbers one less than a square.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 2, 4, 1, 2, 5, 2, 4, 5, 2, 5, 5, 2, 6, 7, 4, 6, 7, 5, 6, 8, 6, 8, 12, 6, 9, 13, 6, 10, 15, 8, 14, 15, 9, 16, 16, 10, 18, 21, 14, 19, 22, 16, 20, 24, 19, 25, 30, 20, 27, 33, 21, 29, 39, 26, 37, 40, 28, 42, 42, 31, 48

Offset: 0

Wouter Meeussen, Oct 22 2005

Also limiting form of the number of representations of n into k positive squares for k decreasing from n to 1, or Table[Count[SumOfSquaresRepresentations[k,n], {a_,}/;a>0], {n,100,100}, {k,100,40,-1}]. (Franklin T. Adams-Watters: replacing k^2 ones by the value k^2 changes the count by k^2-1).

a(n) = A243148(2n,n). - Alois P. Heinz, May 30 2014

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)

Cf. A001156, A078134.

Cf. A005563, A319799.

a111178 = p $ tail a005563_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 02 2014

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<2, 0, b(n, i-1)+
      `if`(i^2>n+1, 0, b(n+1-i^2, i))))
    end:
a:= n-> b(n, isqrt(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, May 30 2014

nn = 100; CoefficientList[Series[Product[1/(1 - x^(k^2 - 1)), {k, 2, nn}], {x, 0, nn}], x] (* corrected by T. D. Noe, Feb 22 2012 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<2, 0, b[n, i-1] + If[i^2>n+1, 0, b[n+1-i^2, i]]]]; a[n_] := b[n, Round[Sqrt[n]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 16 2017, after Alois P. Heinz *)

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

A218494 Number of partitions of n^3 into squares.

Original entry on oeis.org

1, 1, 3, 21, 220, 2846, 41621, 670568, 11570877, 210605770, 3998468431, 78556582448, 1587757499592, 32866068149376, 694307793698105, 14927522659021682, 325895131806047690, 7211436102222542901, 161493494674514291108, 3655277488432342084426

Offset: 0

Reinhard Zumkeller, Oct 31 2012

nonn

			n=2: number of partitions of 8 into parts of {1, 4}:
a(2) = #{4+4, 4+1+1+1+1, 8x1} = 3;
n=3: number of partitions of 27 into parts of  {1, 4, 9, 16, 25}:
a(3) = #{25+1+1, 16+9+1+1, 16+4+4+1, 16+4+5x1, 16+9x1, 9+9+9, 9+9+4+4+1, 9+9+4+5x1, 9+9+9x1, 9+4x4+1+1, 9+3x4+6x1, 9+4+4+10x1, 9+4+14x1, 9+18x1, 6x4+3x1, 5x4+7x1, 4x4+11x1, 3x4+15x1, 4+4+19x1, 4+23x1, 27x1} = 21.

Alois P. Heinz, Table of n, a(n) for n = 0..80

Cf. A000290, A001156, A218495.

a218494 = p (tail a000290_list) . (^ 3) where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
       b(n, i-1)+`if`(i^2>n, 0, b(n-i^2, i)))
    end:
a:= n-> b(n^3, isqrt(n^3)):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 08 2012

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i-1] + If[i^2>n, 0, b[n - i^2, i]]]; a[n_] := b[n^3, Sqrt[n^3] // Floor]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

a(n) = A001156(A000578(n)).
a(n) ~ 3^(-1/2) * (4*Pi)^(-7/6) * Zeta(3/2)^(2/3) * n^(-7/2) * exp(2^(-4/3) * 3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n) [after Hardy & Ramanujan]. - Vaclav Kotesovec, Apr 10 2017
a(n) = [x^(n^3)] Product_{k>=1} 1/(1 - x^(k^2)). - Ilya Gutkovskiy, Jan 29 2018

Extended beyond a(7) by Alois P. Heinz, Nov 08 2012

cp's OEIS Frontend

A131799 Number of partitions of n into parts that are squares or cubes.

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A280129 Expansion of Product_{k>=2} (1 + x^(k^2)).

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A281541 Expansion of Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) / Product_{j>=1} (1 - x^(j^2)).

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A284345 Number of partitions of n into squares dividing n.

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Maple

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A285047 Expansion of Product_{k>=1} 1/(1-x^(k^2))^k.

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A324587 Heinz numbers of integer partitions of n into distinct perfect squares (A033461).

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A351982 Number of integer partitions of n into prime parts with prime multiplicities.

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A092362 Number of partitions of n^2 into squares greater than 1.

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