%I #5 Dec 28 2019 17:03:43
%S 1,1,1,1,1,2,2,1,4,5,5,7,16,16,27,2,61,33,272,27,123,61,1385,27,78,
%T 272,95,123,7936,362
%N Number of non-isomorphic balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.
%C A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
%F For n > 1, a(2^n) = a(prime(n)) = A000111(n - 1).
%e Non-isomorphic representatives of the a(n) multisystems for n = 2, 3, 6, 9, 10, 12 (commas and outer brackets elided):
%e 1 11 {1}{12} {{1}}{{1}{22}} {{1}}{{1}{12}} {{1}}{{1}{23}}
%e {2}{11} {{11}}{{2}{2}} {{11}}{{1}{2}} {{11}}{{2}{3}}
%e {{1}}{{2}{12}} {{1}}{{2}{11}} {{1}}{{2}{13}}
%e {{12}}{{1}{2}} {{12}}{{1}{1}} {{12}}{{1}{3}}
%e {{2}}{{1}{11}} {{2}}{{1}{13}}
%e {{2}}{{3}{11}}
%e {{23}}{{1}{1}}
%Y The non-maximal version is A330666.
%Y The case of constant or strict atoms is A000111.
%Y Labeled versions are A330728, A330665 (prime indices), and A330675 (strongly normal).
%Y Non-isomorphic multiset partitions whose degrees are the prime indices of n are A318285.
%Y Cf. A004114, A005121, A007716, A048816, A141268, A306186, A318846, A318848, A330470, A330474, A330663.
%K nonn,more
%O 1,6
%A _Gus Wiseman_, Dec 28 2019