A087153 -id:A087153 - cp's OEIS frontend

A280204 G.f.: Product_{k>=1} (1+x^(k^2)) / (1-x^k).

1, 2, 3, 5, 9, 14, 21, 31, 45, 65, 92, 127, 175, 239, 322, 430, 572, 753, 985, 1281, 1657, 2131, 2727, 3471, 4401, 5558, 6988, 8751, 10924, 13588, 16846, 20819, 25653, 31518, 38621, 47195, 57530, 69958, 84869, 102723, 124070, 149532, 179852, 215894, 258668

Offset: 0

Vaclav Kotesovec, Dec 28 2016

nonn

Convolution of A033461 and A000041.

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

Cf. A000041, A033461, A052335, A078434, A087153, A117144, A264393, A280224, A280225, A369520, A369570.

nmax=50; CoefficientList[Series[Product[(1+x^(k^2))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(2*n/3) + 2^(-5/4)*3^(1/4)*(sqrt(2)-1)*Zeta(3/2)*n^(1/4) - 3*(sqrt(2)-1)^2*Zeta(3/2)^2/(64*Pi)) / (2^(5/2)*sqrt(3)*n).

A172151 Number of partitions of n into two nonsquares.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 2, 3, 4, 4, 4, 5, 5, 5, 5, 7, 6, 7, 7, 8, 9, 9, 8, 9, 10, 10, 10, 11, 11, 13, 12, 13, 13, 13, 13, 15, 15, 15, 15, 16, 17, 17, 17, 18, 18, 19, 18, 20, 20, 20, 20, 21, 21, 23, 22, 23, 24, 24, 24, 25, 26, 25, 25, 27, 26, 27, 27, 28, 29, 30, 29, 30, 30, 31

Offset: 0

Reinhard Zumkeller, Jan 26 2010

A172152 and A172153 give record values and where they occur: a(A172153(n))=A172152(n) and a(m) < A172152(n) for m < A172153(n).

			a(8) = #{6+2, 5+3} = 2;
a(9) = #{7+2, 6+3} = 2;
a(10) = #{8+2, 7+3, 5+5} = 3;
a(11) = #{8+3, 6+5} = 2;
a(12) = #{10+2, 7+5, 6+6} = 3;
a(13) = #{11+2, 10+3, 8+5, 7+6} = 4.

R. Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000037, A004526, A087153.

a(n)=n\2 - sqrtint(n) + sum(k=sqrtint(n\2)+1,sqrtint(n), issquare(n-k^2)) \\ Charles R Greathouse IV, Aug 28 2016

a(n) = n/2 + O(sqrt(n)). - Charles R Greathouse IV, Aug 28 2016

Typo in b-file link fixed by Reinhard Zumkeller, Feb 10 2010

A285798 Number of partitions of n into parts with an even number of distinct prime divisors.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 5, 5, 6, 7, 8, 8, 11, 11, 14, 16, 19, 19, 25, 26, 31, 34, 40, 41, 51, 53, 62, 68, 80, 85, 103, 107, 124, 135, 157, 166, 195, 205, 235, 256, 294, 311, 362, 383, 437, 472, 535, 568, 652, 695, 786, 847, 954, 1016, 1155, 1231, 1381, 1486, 1662, 1774, 1997, 2130, 2377, 2557, 2846

Offset: 0

Ilya Gutkovskiy, Apr 26 2017

nonn

			a(10) = 3 because we have [10], [6, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A030231, A087153 (number of partitions into parts with an even number of divisors), A285799.

nmax = 70; CoefficientList[Series[Product[1/(1 - Boole[EvenQ[PrimeNu[k]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^A030231(k)).

A369520 Expansion of Product_{k>=1} 1/((1 - x^(k^2))*(1 - x^k)).

Original entry on oeis.org

1, 2, 4, 7, 13, 21, 34, 52, 80, 119, 175, 251, 359, 504, 702, 965, 1320, 1785, 2401, 3200, 4245, 5589, 7324, 9535, 12364, 15944, 20478, 26175, 33338, 42279, 53438, 67283, 84454, 105642, 131764, 163826, 203149, 251185, 309799, 381079, 467666, 572520, 699342, 852314

Offset: 0

Vaclav Kotesovec, Jan 25 2024

nonn

Convolution of A001156 and A000041.

a(n) is the number of pairs (Q(k), P(n-k)), 0<=k<=n, where Q(k) is a partition of k into squares and P(n-k) is a partition of n-k.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (250000 terms)

Cf. A000041, A001156, A087153, A264393, A280204, A369519, A369579.

nmax=50; CoefficientList[Series[Product[1/(1-x^(k^2))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

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)) / (2^(13/4) * 3^(3/4) * sqrt(Pi) * n^(5/4)).

A087154 Number of partitions of n into distinct nonsquares.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 1, 2, 3, 2, 4, 4, 4, 7, 6, 9, 9, 11, 14, 14, 19, 21, 23, 29, 31, 36, 43, 46, 56, 62, 70, 81, 91, 103, 117, 132, 148, 167, 188, 211, 237, 266, 297, 332, 371, 414, 461, 515, 571, 634, 708, 780, 870, 963, 1062, 1180, 1300, 1436, 1588, 1747, 1929, 2123

Offset: 0

Reinhard Zumkeller, Aug 21 2003

nonn

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

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

Cf. A087153, A033461, A000041, A000037.

Cf. A225045, A280264.

a087154 = p a000037_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Apr 25 2013

nmax = 100; CoefficientList[Series[Product[(1 + x^k)/(1 + x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 29 2016 *)

G.f.: Product_{m>0} (1+x^m)/(1+x^(m^2)). - Vladeta Jovovic, Jul 31 2004

a(n) ~ exp(Pi*sqrt(n/3) - 3^(1/4) * (sqrt(2)-1) * Zeta(3/2) * n^(1/4) / 2 - 3*(sqrt(2)-1)^2 * Zeta(3/2)^2 / (32*Pi)) / (2^(3/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Dec 30 2016

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

A225044 Number of partitions of n into non-triangular numbers, cf. A014132.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 2, 2, 4, 4, 5, 6, 9, 10, 13, 14, 21, 22, 30, 32, 44, 48, 62, 69, 89, 100, 124, 141, 173, 198, 239, 274, 330, 377, 450, 514, 611, 697, 823, 939, 1104, 1258, 1470, 1676, 1950, 2220, 2572, 2927, 3381, 3841, 4420, 5019, 5759, 6529, 7470, 8460

Offset: 0

Reinhard Zumkeller, Apr 25 2013

nonn

			a(10) = #{8+2, 5+5, 4+4+2, 4+2+2+2, 2+2+2+2+2} = 5;
a(11) = #{11, 9+2, 7+4, 7+2+2, 5+4+2, 5+2+2+2} = 6;
a(12) = #{12, 8+4, 8+2+2, 7+5, 5+5+2, 4+4+4, 4+4+2+2, 4+2+2+2+2, 6x2} = 9;
a(13) = #{13, 11+2, 9+4, 9+2+2, 8+5, 7+4+2, 7+2+2+2, 5+4+4, 5+4+2+2, 5+2+2+2+2} = 10.

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

Cf. A000217, A225045, A087153.

Column k=0 of A263234.

a225044 = p a014132_list 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, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i>n or issqr(8*i+1), 0, b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 13 2015

t = Table[n (n + 1)/2, {n, 1, 200}] ; p[n_] := IntegerPartitions[n, All, Complement[Range@n, t]]; Table[p[n], {n, 0, 12}] (*shows partitions*)
a[n_] := Length@p@n; a /@ Range[0, 80]
(* Clark Kimberling, Mar 09 2014 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n || IntegerQ @ Sqrt[8*i + 1], 0, b[n - i, i]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

G.f.: Product_{k>=1} (1 - x^(k*(k+1)/2))/(1 - x^k). - Ilya Gutkovskiy, Dec 30 2016

a(n) ~ sqrt(2) * exp(Pi*sqrt(2*n/3) - Zeta(3/2) * (3*n/2)^(1/4) - 3*Zeta(3/2)^2 / (16*Pi)) / sqrt(n). - Vaclav Kotesovec, Jan 01 2017

A286219 Number of partitions of n into parts with an even number of prime divisors (counted with multiplicity).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 7, 9, 10, 13, 15, 19, 20, 25, 28, 34, 38, 46, 50, 61, 69, 81, 89, 105, 116, 137, 152, 175, 194, 226, 250, 288, 318, 363, 403, 462, 508, 577, 637, 721, 796, 900, 988, 1113, 1228, 1378, 1515, 1696, 1860, 2080, 2287, 2546, 2791, 3106, 3402, 3779

Offset: 0

Ilya Gutkovskiy, May 04 2017

nonn

			a(8) = 4 because we have [6, 1, 1], [4, 4], [4, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Prime Factor
Index entries for related partition-counting sequences

Cf. A028260, A087153, A285798, A286218.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      `if`(bigomega(d)::odd, 0, d), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, May 04 2017

nmax = 60; CoefficientList[Series[Product[1/(1 - Boole[EvenQ[PrimeOmega[k]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^A028260(k)).

A285309 Sum of nonsquare divisors of n.

Original entry on oeis.org

0, 2, 3, 2, 5, 11, 7, 10, 3, 17, 11, 23, 13, 23, 23, 10, 17, 29, 19, 37, 31, 35, 23, 55, 5, 41, 30, 51, 29, 71, 31, 42, 47, 53, 47, 41, 37, 59, 55, 85, 41, 95, 43, 79, 68, 71, 47, 103, 7, 67, 71, 93, 53, 110, 71, 115, 79, 89, 59, 163, 61, 95, 94, 42, 83, 143, 67, 121, 95, 143, 71, 145, 73, 113, 98

Offset: 1

Ilya Gutkovskiy, Apr 16 2017

nonn

			a(6) = 11 because 6 has 4 divisors {1, 2, 3, 6} among which 3 are nonsquares {2, 3, 6} therefore 2 + 3 + 6 = 11.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors

Cf. A000037, A000203, A005117, A035316, A056595, A087153.

Table[DivisorSum[n, # &, Mod[DivisorSigma[0, #], 2] == 0 &], {n, 1, 75}]
nmax = 75; Rest[CoefficientList[Series[Sum[(k + Floor[1/2 + Sqrt[k]]) x^(k + Floor[1/2 + Sqrt[k]])/(1 - x^(k + Floor[1/2 + Sqrt[k]])), {k, 1, nmax}], {x, 0, nmax}], x]]
Array[DivisorSum[#, # &, ! IntegerQ@ Sqrt@ # &] &, 75] (* Michael De Vlieger, Nov 23 2017 *)

a(n) = sumdiv(n, d, if (!issquare(d), d)); \\ Michel Marcus, Apr 17 2017

import gmpy
from sympy import divisors
def a(n): return sum([d for d in divisors(n) if gmpy.is_square(d)==0]) # Indranil Ghosh, Apr 18 2017

G.f.: Sum_{k>=1} A000037(k)*x^A000037(k)/(1 - x^A000037(k)).
a(n) = A000203(n) - A035316(n).
a(A005117(n)) = A000203(A005117(n)) - 1.
a(p^(2*k-1)) = a(p^(2*k)) = p*(p^(2*k) - 1)/(p^2 - 1) for p is a prime and k >= 1.

A369519 Expansion of Product_{k>=1} 1/((1 - x^(k^2))*(1 - x^(k^3))).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 16, 21, 26, 31, 38, 46, 54, 62, 74, 88, 103, 118, 137, 158, 180, 202, 230, 263, 298, 335, 378, 426, 476, 528, 589, 658, 732, 810, 900, 998, 1101, 1208, 1330, 1465, 1608, 1760, 1930, 2116, 2310, 2513, 2738, 2985, 3246, 3521, 3826, 4156

Offset: 0

Vaclav Kotesovec, Jan 25 2024

nonn

Convolution of A001156 and A003108.

a(n) is the number of pairs (Q(k), P(n-k)), 0<=k<=n, where Q(k) is a partition of k into squares and P(n-k) is a partition of n-k into cubes.

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

Cf. A001156, A003108.

Cf. A087153, A131799, A264393, A369520, A369579.

nmax=100; CoefficientList[Series[Product[1/(1-x^(k^2))/(1-x^(k^3)), {k, 1, nmax^(1/2)}], {x, 0, nmax}], x]

a(n) ~ zeta(3/2) * exp(3 * Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3) / 2^(4/3) + 2^(4/9) * Gamma(1/3) * zeta(4/3) * n^(2/9) / (3 * Pi^(1/9) * zeta(3/2)^(2/9)) - 4*2^(2/9) * Gamma(1/3)^2 * zeta(4/3)^2 * n^(1/9) / (243 * Pi^(5/9) * zeta(3/2)^(10/9)) + 16*Gamma(1/3)^3 * zeta(4/3)^3 / (6561 * Pi * zeta(3/2)^2)) / (16 * sqrt(6) * Pi^(5/2) * n^(3/2)) * (1 + (13*2^(7/9) * Gamma(1/3) * zeta(4/3) / (81 * Pi^(4/9) * zeta(3/2)^(8/9)) + 832*2^(7/9) * Gamma(1/3)^4 * zeta(4/3)^4 / (1594323 * Pi^(13/9) * zeta(3/2)^(26/9))) / n^(1/9) + (692224 * 2^(5/9) * Gamma(1/3)^8 * zeta(4/3)^8 / (2541865828329 * Pi^(26/9) * zeta(3/2)^(52/9)) - 128 * 2^(5/9) * Gamma(1/3)^5 * zeta(4/3)^5 / (4782969 * Pi^(17/9) * zeta(3/2)^(34/9)) + 65*2^(5/9) * Gamma(1/3)^2 * zeta(4/3)^2 / (2187*Pi^(8/9) * zeta(3/2)^(16/9)))/n^(2/9)).

A303947 Number of partitions of n into at most 1 copy of 2^2, 2 copies of 3^2, 3 copies of 4^2, ... .

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 0, 1, 0, 2, 0, 2, 0, 2, 0, 1, 2, 0, 1, 0, 3, 0, 1, 1, 2, 3, 0, 2, 1, 4, 0, 1, 2, 2, 2, 0, 4, 1, 2, 1, 3, 3, 1, 3, 1, 5, 1, 3, 2, 4, 3, 1, 4, 2, 4, 1, 5, 3, 3, 4, 4, 6, 2, 4

Offset: 0

Seiichi Manyama, May 03 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A033461, A087153, A303942.

G.f.: Product_{k>=2} (1-x^(k^3))/(1-x^(k^2)).

cp's OEIS Frontend

A280204 G.f.: Product_{k>=1} (1+x^(k^2)) / (1-x^k).

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A172151 Number of partitions of n into two nonsquares.

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PARI

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A285798 Number of partitions of n into parts with an even number of distinct prime divisors.

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A369520 Expansion of Product_{k>=1} 1/((1 - x^(k^2))*(1 - x^k)).

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A087154 Number of partitions of n into distinct nonsquares.

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Haskell

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A225044 Number of partitions of n into non-triangular numbers, cf. A014132.

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Haskell

Maple

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A286219 Number of partitions of n into parts with an even number of prime divisors (counted with multiplicity).

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Maple

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A285309 Sum of nonsquare divisors of n.

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