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A181887 a(0) = 0, and for n > 0, a(n) = A002956(n) - A000041(n).

%I A181887 #12 Jun 02 2025 03:13:41
%S A181887 0,0,0,1,2,8,9,33,43,89,124,292,290,726,839,1318,1904,3616,3653,7446,
%T A181887 7620,12175,16474,27907,26490,47651,56922,80410,93525,160402,146944,
%U A181887 273510,286942,395776,495852,659747,690842
%N A181887 a(0) = 0, and for n > 0, a(n) = A002956(n) - A000041(n).
%C A181887 A002956 can be thought of as a modular arithmetic version of the partition numbers (A000041). The number of "modulo n" partitions of n is the number of multisets of integers ranging from 1 to n, such that the sum of members of the multiset is congruent to 0 mod n, and no submultiset exists whose members sum to 0 mod n. Therefore, a(n) is the number of "modulo n" partitions which are not ordinary partitions of n.
%H A181887 Finklea, Moore, Ponomarenko and Turner, <a href="http://www-rohan.sdsu.edu/~vadim/fmpt1b-revised.pdf">Invariant Polynomials and Minimal Zero Sequences</a>
%e A181887 The multisets counted by A002956(5) but not by A000041(5) are
%e A181887 ..{1,3,3,3}
%e A181887 ..{2,2,2,2,2}
%e A181887 ..{2,2,2,4}
%e A181887 ..{2,4,4}
%e A181887 ..{3,3,3,3,3}
%e A181887 ..{3,4,4,4}
%e A181887 ..{3,3,4}
%e A181887 ..{4,4,4,4,4}
%e A181887 So a(5) = 8.
%Y A181887 Cf. A000041, A002956, A082641
%K A181887 nonn
%O A181887 0,5
%A A181887 _Andrew Weimholt_, Feb 01 2011