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A385374 a(n) is the number of partitions of n into tau(n) distinct parts.

%I A385374 #12 Jul 11 2025 15:35:08
%S A385374 1,0,1,0,2,0,3,0,3,1,5,0,6,5,6,1,8,0,9,0,27,34,11,0,40,64,72,14,14,0,
%T A385374 15,44,150,169,185,0,18,249,270,5,20,11,21,454,532,478,23,0,176,1057,
%U A385374 672,1360,26,288,864,434,972,1033,29,0,30,1285,4494,4011,1495
%N A385374 a(n) is the number of partitions of n into tau(n) distinct parts.
%H A385374 Felix Huber, <a href="/A385374/b385374.txt">Table of n, a(n) for n = 1..4000</a>
%e A385374 a(14) = 5 because there are 5 partitions of 14 into tau(14) = 4 distinct parts: [1, 2, 3, 8], [1, 2, 4, 7], [1, 2, 5, 6], [1, 3, 4, 6], [2, 3, 4, 5].
%p A385374 b:= proc(n,i,k)
%p A385374     option remember;
%p A385374     if n=0 and k=0 then
%p A385374         return 1
%p A385374     elif n=0 or k=0 or i<1 then
%p A385374         return 0
%p A385374     elif i<=n then
%p A385374         return b(n,i-1,k)+b(n-i,i-1,k-1)
%p A385374     else
%p A385374         return b(n,i-1,k)
%p A385374     fi;
%p A385374 end proc:
%p A385374 A385374:=n->b(n,n,NumberTheory:-tau(n));
%p A385374 seq(A385374(n),n=1..65);
%t A385374 a[n_]:=Length[Select[Union/@IntegerPartitions[n,{DivisorSigma[0,n]}],Length[#]==DivisorSigma[0,n]&]];Array[a,65] (* _James C. McMahon_, Jul 11 2025 *)
%Y A385374 Subsequence of A060016.
%Y A385374 Cf. A000005, A385375.
%K A385374 nonn,look
%O A385374 1,5
%A A385374 _Felix Huber_, Jul 06 2025