cp's OEIS Frontend

A330664 Number of non-isomorphic balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.

%I A330664 #5 Dec 28 2019 17:03:43
%S A330664 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 A330664 272,95,123,7936,362
%N A330664 Number of non-isomorphic balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.
%C A330664 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 A330664 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 A330664 For n > 1, a(2^n) = a(prime(n)) = A000111(n - 1).
%e A330664 Non-isomorphic representatives of the a(n) multisystems for n = 2, 3, 6, 9, 10, 12 (commas and outer brackets elided):
%e A330664   1  11  {1}{12}  {{1}}{{1}{22}}  {{1}}{{1}{12}}  {{1}}{{1}{23}}
%e A330664          {2}{11}  {{11}}{{2}{2}}  {{11}}{{1}{2}}  {{11}}{{2}{3}}
%e A330664                   {{1}}{{2}{12}}  {{1}}{{2}{11}}  {{1}}{{2}{13}}
%e A330664                   {{12}}{{1}{2}}  {{12}}{{1}{1}}  {{12}}{{1}{3}}
%e A330664                                   {{2}}{{1}{11}}  {{2}}{{1}{13}}
%e A330664                                                   {{2}}{{3}{11}}
%e A330664                                                   {{23}}{{1}{1}}
%Y A330664 The non-maximal version is A330666.
%Y A330664 The case of constant or strict atoms is A000111.
%Y A330664 Labeled versions are A330728, A330665 (prime indices), and A330675 (strongly normal).
%Y A330664 Non-isomorphic multiset partitions whose degrees are the prime indices of n are A318285.
%Y A330664 Cf. A004114, A005121, A007716, A048816, A141268, A306186, A318846, A318848, A330470, A330474, A330663.
%K A330664 nonn,more
%O A330664 1,6
%A A330664 _Gus Wiseman_, Dec 28 2019