cp's OEIS Frontend

A330666 Number of non-isomorphic balanced reduced multisystems whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.

%I A330666 #5 Dec 31 2019 08:24:07
%S A330666 1,1,1,1,2,3,6,2,10,11,20,15,90,51,80,6,468,93,2910,80,521,277,20644,
%T A330666 80,334,1761,393,521,165874,1374
%N A330666 Number of non-isomorphic balanced reduced multisystems whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.
%C A330666 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 A330666 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 A330666 a(2^n) = a(prime(n)) = A318813(n).
%e A330666 Non-isomorphic representatives of the a(2) = 1 through a(9) = 10 multisystems (commas and outer brackets elided):
%e A330666     1  11  12  111      112      1111            123      1122
%e A330666                {1}{11}  {1}{12}  {1}{111}        {1}{23}  {1}{122}
%e A330666                         {2}{11}  {11}{11}                 {11}{22}
%e A330666                                  {1}{1}{11}               {12}{12}
%e A330666                                  {{1}}{{1}{11}}           {1}{1}{22}
%e A330666                                  {{11}}{{1}{1}}           {1}{2}{12}
%e A330666                                                           {{1}}{{1}{22}}
%e A330666                                                           {{11}}{{2}{2}}
%e A330666                                                           {{1}}{{2}{12}}
%e A330666                                                           {{12}}{{1}{2}}
%e A330666 Non-isomorphic representatives of the a(12) = 15 multisystems:
%e A330666   {1,1,2,3}
%e A330666   {{1},{1,2,3}}
%e A330666   {{1,1},{2,3}}
%e A330666   {{1,2},{1,3}}
%e A330666   {{2},{1,1,3}}
%e A330666   {{1},{1},{2,3}}
%e A330666   {{1},{2},{1,3}}
%e A330666   {{2},{3},{1,1}}
%e A330666   {{{1}},{{1},{2,3}}}
%e A330666   {{{1,1}},{{2},{3}}}
%e A330666   {{{1}},{{2},{1,3}}}
%e A330666   {{{1,2}},{{1},{3}}}
%e A330666   {{{2}},{{1},{1,3}}}
%e A330666   {{{2}},{{3},{1,1}}}
%e A330666   {{{2,3}},{{1},{1}}}
%Y A330666 The labeled version is A318846.
%Y A330666 The maximum-depth version is A330664.
%Y A330666 Unlabeled balanced reduced multisystems by weight are A330474.
%Y A330666 The case of constant or strict atoms is A318813.
%Y A330666 Cf. A000669, A005121, A007716, A048816, A141268, A306186, A317791, A318812, A318849, A330470, A330475, A330655, A330728.
%K A330666 nonn,more
%O A330666 1,5
%A A330666 _Gus Wiseman_, Dec 30 2019