cp's OEIS Frontend

A334652 Number of integer partitions of n with at least two parts, each greater than 1 and with the same multiplicity.

%I A334652 #16 Jun 27 2025 22:24:13
%S A334652 0,0,0,0,1,1,3,2,4,5,7,6,12,9,15,17,21,20,33,28,43,44,55,55,81,77,99,
%T A334652 108,135,136,184,180,230,246,294,316,398,403,489,532,637,668,816,852,
%U A334652 1019,1107,1275,1370,1637,1727,2016,2185,2518,2701,3152,3370,3884,4200,4774,5154,5953
%N A334652 Number of integer partitions of n with at least two parts, each greater than 1 and with the same multiplicity.
%C A334652 All parts are greater than 1, there are at least two parts, and each part size has the same multiplicity.
%C A334652 This sequence was inspired by a post of Ali Sada, May 07 2020 on the seqfan mailing list.
%H A334652 Andrew Howroyd, <a href="/A334652/b334652.txt">Table of n, a(n) for n = 0..1000</a>
%F A334652 a(n) = -1 + Sum_{d|n} A025147(d) for n > 1. - _Andrew Howroyd_, May 07 2020
%e A334652 The a(4) = 1 partition is 2 + 2.
%e A334652 The a(7) = 2 partitions are 2 + 5 and 3 + 4. Each part has multiplicity 1.
%t A334652 Table[Length@Select[IntegerPartitions[n], Min[#] > 1 && Length[#] > 1 && (Length[Union[Length /@ Split[Sort[#]]]] == 1) &], {n, 0, 20}]
%o A334652 (PARI) \\ here b(n) is A025147.
%o A334652 b(n)={my(A=O(x*x^n)); polcoef(eta(x^2 + A) / eta(x + A) / (1 + x), n)}
%o A334652 a(n)={if(n<=1, 0, sumdiv(n, d, b(d)) - 1)} \\ _Andrew Howroyd_, May 07 2020
%Y A334652 Cf. A025147, A083751, A334653.
%K A334652 nonn
%O A334652 0,7
%A A334652 _Olivier Gérard_, May 07 2020