A083214 -id:A083214 - cp's OEIS frontend

A087183 Partition numbers of the form 3*k.

3, 15, 30, 42, 135, 231, 297, 627, 792, 1002, 1575, 2436, 5604, 8349, 10143, 14883, 31185, 37338, 44583, 63261, 105558, 147273, 239943, 281589, 329931, 614154, 1121505, 1505499, 3087735, 4087968, 4697205, 8118264, 15796476, 44108109

Offset: 1

Reinhard Zumkeller, Aug 23 2003

nonn

The numbers m such that 3 divides A000041(m) are given in A083214. Klarreich writes: no one has proved whether there are infinitely many partition numbers divisible by 3, although it is known that there are infinitely many partition numbers divisible by 2. - Jonathan Vos Post, Jul 31 2008

Intersection of A008585 and A000041. - Reinhard Zumkeller, Nov 03 2009

Erica Klarreich, Pieces of numbers: a proof brings closure to a dramatic tale of partitions and primes, Science News, Jun 18 2005.

Paul Tek, Table of n, a(n) for n = 1..10000
Erica Klarreich, Pieces of numbers: a proof brings closure to a dramatic tale of partitions and primes, Science News, Jun 18 2005.
Eric Weisstein's World of Mathematics, Partition Function.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.

Cf. A000041, A008585, A068907, A052001, A052003, A079978, A083214, A087180, A087184, A087185, A167392, A213365.

Select[PartitionsP@Range[120], Divisible[#, 3] &] (* Vladimir Reshetnikov, Nov 05 2015 *)

for(n=9, 1e3, t=numbpart(n); if(t%3, , print1(t", "))) \\ Charles R Greathouse IV, May 08 2013

A079978(a(n))*A167392(a(n)) = 1. - Reinhard Zumkeller, Nov 03 2009
a(n) = 3*A213365(n). - Omar E. Pol, May 08 2013
a(n) = A000041(A083214(n)). - Amiram Eldar, May 22 2025

A243935 Numbers m such that 5 divides A000041(m).

Original entry on oeis.org

4, 7, 9, 14, 18, 19, 23, 24, 27, 29, 34, 38, 39, 44, 49, 54, 58, 59, 61, 64, 66, 68, 69, 71, 74, 79, 82, 84, 89, 94, 97, 99, 103, 104, 109, 114, 119, 120, 124, 127, 128, 129, 130, 134, 136, 139, 140, 142, 144, 149, 154, 159, 163, 164, 165, 169, 170, 173, 174

Offset: 1

Bruno Berselli, Jun 15 2014

Bruno Berselli, Table of n, a(n) for n = 1..1000

Numbers m such that k divides A000041(m), where k is prime: A001560 (k=2), A083214 (k=3), this sequence (k=5), A243936 (k=7), A027827 (k=11), A071750 (k=13). For k composite: A237278 (k=4), A035700 (k=12).

[n: n in [1..200] | IsZero(NumberOfPartitions(n) mod 5)];

Select[Range[200], Mod[PartitionsP[#], 5] == 0 &]

is(n)=numbpart(n)%5==0 \\ Charles R Greathouse IV, Apr 08 2015

# From Peter Luschny in A000041
@CachedFunction
def A000041(n):
    if n == 0: return 1
    S = 0; J = n-1; k = 2
    while 0 <= J:
        T = A000041(J)
        S = S+T if is_odd(k//2) else S-T
        J -= k if is_odd(k) else k//2
        k += 1
    return S
[n for n in (0..200) if mod(A000041(n),5) == 0]

A237276 Numbers k such that A000041(k) == 1 (mod 3).

Original entry on oeis.org

0, 1, 5, 8, 18, 19, 23, 27, 28, 34, 36, 37, 44, 45, 50, 54, 55, 59, 62, 64, 72, 73, 77, 81, 82, 86, 89, 91, 95, 98, 99, 100, 110, 112, 113, 116, 117, 118, 119, 122, 128, 134, 137, 139, 140, 143, 146, 148, 149, 150, 152, 154, 155, 157, 158, 161, 166, 168, 170

Offset: 1

Clark Kimberling, Feb 05 2014

The set of positive integers is partitioned by A083214, A237276, and A237277.

			A000041(8) = 22 == 1 (mod 3).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Clark Kimberling)

Cf. A000041, A083214, A237276, A237277.

f[n_, k_] := Select[Range[0, 250], Mod[PartitionsP[#], n] == k &]
Table[f[3, k], {k, 0, 2}] (* A083214, A237276, A237277 *)
Table[f[4, k], {k, 0, 3}] (* A237278-A237281 *)

a(1)=0 inserted by Amiram Eldar, May 22 2025

A237277 Numbers k such that A000041(k) == 2 (mod 3).

Original entry on oeis.org

2, 4, 6, 11, 12, 13, 15, 25, 29, 31, 38, 42, 47, 49, 56, 58, 60, 65, 66, 67, 69, 74, 76, 78, 79, 83, 84, 85, 87, 90, 92, 93, 94, 96, 101, 103, 105, 108, 109, 114, 120, 121, 123, 126, 127, 130, 131, 132, 135, 136, 141, 144, 145, 153, 156, 159, 165, 167, 171

Offset: 1

Clark Kimberling, Feb 05 2014

The set of positive integers is partitioned by A083214, A237276, and A237277.

			A000041(6) = 11 == 2 (mod 3).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000041, A083214, A237276.

f[n_, k_] := Select[Range[250], Mod[PartitionsP[#], n] == k &]
Table[f[3, k], {k, 0, 2}] (* A083214, A237276, A237277 *)
Table[f[4, k], {k, 0, 3}] (* A237278-A237281 *)

A243936 Numbers m such that 7 divides A000041(m).

Original entry on oeis.org

5, 10, 11, 12, 16, 18, 19, 24, 26, 27, 33, 37, 39, 40, 41, 47, 48, 52, 53, 54, 55, 61, 68, 75, 76, 82, 83, 89, 96, 97, 103, 110, 111, 117, 124, 125, 131, 138, 140, 145, 147, 152, 159, 166, 170, 173, 177, 180, 187, 191, 194, 201, 208, 213, 215, 222, 225, 229, 232

Offset: 1

Bruno Berselli, Jun 15 2014

Bruno Berselli, Table of n, a(n) for n = 1..1000

Numbers m such that k divides A000041(m), where k is prime: A001560 (k=2), A083214 (k=3), A243935 (k=5), this sequence (k=7), A027827 (k=11), A071750 (k=13). For k composite: A237278 (k=4), A035700 (k=12).

[n: n in [1..250] | IsZero(NumberOfPartitions(n) mod 7)];

Select[Range[250], Mod[PartitionsP[#], 7] == 0 &]

is(n)=numbpart(n)%7==0 \\ Charles R Greathouse IV, Apr 08 2015

# From Peter Luschny in A000041
@CachedFunction
def A000041(n):
    if n == 0: return 1
    S = 0; J = n-1; k = 2
    while 0 <= J:
        T = A000041(J)
        S = S+T if is_odd(k//2) else S-T
        J -= k if is_odd(k) else k//2
        k += 1
    return S
[n for n in (0..250) if mod(A000041(n),7) == 0]

A127174 Numbers n of the form 3k such that partition number of n is also of the form 3k.

Original entry on oeis.org

3, 9, 21, 24, 30, 33, 39, 48, 51, 57, 63, 75, 102, 111, 129, 138, 147, 162, 180, 189, 195, 198, 207, 222, 225, 231, 240, 249, 267, 270, 288, 297, 330, 336, 339, 342, 348, 351, 354, 357, 363, 369, 372, 381, 396, 399, 402, 405, 411, 429, 432, 465, 468, 477, 480

Offset: 1

Zak Seidov, Apr 05 2007

nonn

Subset of A083214. Or, intersection of A083214 and A008585.

Cf. A000041, A008585, A083214, A087183.

with(combinat): a:=proc(k): if numbpart(3*k) mod 3 = 0 then 3*k else fi end: seq(a(n),n=1..200); # Emeric Deutsch, Apr 16 2007

Select[Range[3,600,3],Mod[PartitionsP[ # ],3]==0&]

cp's OEIS Frontend

A087183 Partition numbers of the form 3*k.

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A243935 Numbers m such that 5 divides A000041(m).

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A237276 Numbers k such that A000041(k) == 1 (mod 3).

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A237277 Numbers k such that A000041(k) == 2 (mod 3).

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A243936 Numbers m such that 7 divides A000041(m).

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Magma

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Sage

A127174 Numbers n of the form 3*k such that partition number of n is also of the form 3*k.

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A127174 Numbers n of the form 3k such that partition number of n is also of the form 3k.