%I #5 Dec 31 2019 08:24:07
%S 1,1,1,1,2,3,6,2,10,11,20,15,90,51,80,6,468,93,2910,80,521,277,20644,
%T 80,334,1761,393,521,165874,1374
%N Number of non-isomorphic balanced reduced multisystems 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 a(2^n) = a(prime(n)) = A318813(n).
%e Non-isomorphic representatives of the a(2) = 1 through a(9) = 10 multisystems (commas and outer brackets elided):
%e 1 11 12 111 112 1111 123 1122
%e {1}{11} {1}{12} {1}{111} {1}{23} {1}{122}
%e {2}{11} {11}{11} {11}{22}
%e {1}{1}{11} {12}{12}
%e {{1}}{{1}{11}} {1}{1}{22}
%e {{11}}{{1}{1}} {1}{2}{12}
%e {{1}}{{1}{22}}
%e {{11}}{{2}{2}}
%e {{1}}{{2}{12}}
%e {{12}}{{1}{2}}
%e Non-isomorphic representatives of the a(12) = 15 multisystems:
%e {1,1,2,3}
%e {{1},{1,2,3}}
%e {{1,1},{2,3}}
%e {{1,2},{1,3}}
%e {{2},{1,1,3}}
%e {{1},{1},{2,3}}
%e {{1},{2},{1,3}}
%e {{2},{3},{1,1}}
%e {{{1}},{{1},{2,3}}}
%e {{{1,1}},{{2},{3}}}
%e {{{1}},{{2},{1,3}}}
%e {{{1,2}},{{1},{3}}}
%e {{{2}},{{1},{1,3}}}
%e {{{2}},{{3},{1,1}}}
%e {{{2,3}},{{1},{1}}}
%Y The labeled version is A318846.
%Y The maximum-depth version is A330664.
%Y Unlabeled balanced reduced multisystems by weight are A330474.
%Y The case of constant or strict atoms is A318813.
%Y Cf. A000669, A005121, A007716, A048816, A141268, A306186, A317791, A318812, A318849, A330470, A330475, A330655, A330728.
%K nonn,more
%O 1,5
%A _Gus Wiseman_, Dec 30 2019