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