cp's OEIS Frontend

A334653 Number of integer partitions of n with at least two parts, each greater than 1, at least two kinds of parts and all with the same multiplicity.

%I A334653 #22 Jun 27 2025 22:24:26
%S A334653 0,0,0,0,0,1,1,2,2,4,5,6,8,9,13,15,18,20,29,28,39,42,53,55,75,76,97,
%T A334653 106,131,136,178,180,226,244,292,314,391,403,487,530,631,668,810,852,
%U A334653 1015,1103,1273,1370,1629,1726,2012,2183,2514,2701,3146,3368,3878,4198
%N A334653 Number of integer partitions of n with at least two parts, each greater than 1, at least two kinds of parts and all with the same multiplicity.
%C A334653 All parts are greater than 1, there is more than one part, and each part size has the same multiplicity.
%C A334653 This sequence was inspired by a post of Ali Sada, May 07 2020, on the seqfan mailing list.
%H A334653 Andrew Howroyd, <a href="/A334653/b334653.txt">Table of n, a(n) for n = 0..1000</a>
%F A334653 a(n) = 1 + Sum_{d|n} (A025147(d) - 1) for n > 0. - _Andrew Howroyd_, May 07 2020
%e A334653 The a(10) = 5 partitions are 2 + 8, 3 + 7, 4 + 6, 2 + 3 + 5 and 2 + 2 + 3 + 3.
%t A334653 Table[Length@Select[IntegerPartitions[n], Min[#] > 1 && Length[#] > 1 && Length[Union[#]] > 1 && (Length[Union[Length /@ Split[Sort[#]]]] == 1) &], {n, 0, 40}]
%o A334653 (PARI) \\ here b(n) is A025147.
%o A334653 b(n)={my(A=O(x*x^n)); polcoef(eta(x^2 + A) / eta(x + A) / (1 + x), n)}
%o A334653 a(n)={if(n<1, 0, 1 + sumdiv(n, d, b(d)-1))} \\ _Andrew Howroyd_, May 07 2020
%Y A334653 Cf. A025147, A083751, A334652.
%K A334653 nonn
%O A334653 0,8
%A A334653 _Olivier Gérard_, May 07 2020