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

1, 1, 1, 1, 2, 3, 6, 2, 10, 11, 20, 15, 90, 51, 80, 6, 468, 93, 2910, 80, 521, 277, 20644, 80, 334, 1761, 393, 521, 165874, 1374

Offset: 1

Author

Gus Wiseman, Dec 30 2019

Keywords

• nonn
• more

Comments

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.

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}.

Examples

			Non-isomorphic representatives of the a(2) = 1 through a(9) = 10 multisystems (commas and outer brackets elided):
    1  11  12  111      112      1111            123      1122
               {1}{11}  {1}{12}  {1}{111}        {1}{23}  {1}{122}
                        {2}{11}  {11}{11}                 {11}{22}
                                 {1}{1}{11}               {12}{12}
                                 {{1}}{{1}{11}}           {1}{1}{22}
                                 {{11}}{{1}{1}}           {1}{2}{12}
                                                          {{1}}{{1}{22}}
                                                          {{11}}{{2}{2}}
                                                          {{1}}{{2}{12}}
                                                          {{12}}{{1}{2}}
Non-isomorphic representatives of the a(12) = 15 multisystems:
  {1,1,2,3}
  {{1},{1,2,3}}
  {{1,1},{2,3}}
  {{1,2},{1,3}}
  {{2},{1,1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{2},{3},{1,1}}
  {{{1}},{{1},{2,3}}}
  {{{1,1}},{{2},{3}}}
  {{{1}},{{2},{1,3}}}
  {{{1,2}},{{1},{3}}}
  {{{2}},{{1},{1,3}}}
  {{{2}},{{3},{1,1}}}
  {{{2,3}},{{1},{1}}}
		

Crossrefs

The labeled version is A318846.

The maximum-depth version is A330664.

Unlabeled balanced reduced multisystems by weight are A330474.

The case of constant or strict atoms is A318813.

Cf. A000669, A005121, A007716, A048816, A141268, A306186, A317791, A318812, A318849, A330470, A330475, A330655, A330728.

Formula

a(2^n) = a(prime(n)) = A318813(n).