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

Author

Reinhard Zumkeller, Aug 23 2003

Keywords

• nonn

Comments

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

References

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

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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