A321270 Number of connected multiset partitions of a multiset whose multiplicities are the prime indices of n.

1, 1, 2, 1, 3, 2, 5, 1, 5, 4, 7, 3, 11, 7, 10, 1, 15, 9, 22, 7, 19, 12, 30, 5, 22, 19, 28, 14, 42, 22, 56, 1, 33, 30, 42, 20, 77, 45

Offset

1,3

Comments

This multiset (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}.

Links

Table of n, a(n) for n=1..38.

Example

The a(2) = 1 through a(12) = 3 connected multiset partitions:
  {{1}}  {{11}}    {{12}}  {{111}}      {{112}}    {{1111}}
         {{1}{1}}          {{1}{11}}    {{1}{12}}  {{1}{111}}
                           {{1}{1}{1}}             {{11}{11}}
                                                   {{1}{1}{11}}
                                                   {{1}{1}{1}{1}}
.
  {{123}}  {{1122}}      {{1112}}      {{11111}}          {{1123}}
           {{1}{122}}    {{1}{112}}    {{1}{1111}}        {{1}{123}}
           {{12}{12}}    {{11}{12}}    {{11}{111}}        {{12}{13}}
           {{2}{112}}    {{1}{1}{12}}  {{1}{1}{111}}
           {{1}{2}{12}}                {{1}{11}{11}}
                                       {{1}{1}{1}{11}}
                                       {{1}{1}{1}{1}{1}}
The a(18) = 9, a(27) = 28, and a(36) = 20 connected multiset partitions of {1,1,2,2,3}, {1,1,2,2,3,3}, and {1,1,2,2,3,4} respectively:
  {{1,1,2,2,3}}      {{1,1,2,2,3,3}}        {{1,1,2,2,3,4}}
  {{1},{1,2,2,3}}    {{1},{1,2,2,3,3}}      {{1},{1,2,2,3,4}}
  {{1,2},{1,2,3}}    {{1,1,2},{2,3,3}}      {{1,1,2},{2,3,4}}
  {{1,3},{1,2,2}}    {{1,1,3},{2,2,3}}      {{1,2},{1,2,3,4}}
  {{2},{1,1,2,3}}    {{1,2},{1,2,3,3}}      {{1,2,2},{1,3,4}}
  {{2,3},{1,1,2}}    {{1,2,2},{1,3,3}}      {{1,2,3},{1,2,4}}
  {{1},{1,2},{2,3}}  {{1,2,3},{1,2,3}}      {{1,3},{1,2,2,4}}
  {{1},{2},{1,2,3}}  {{1,3},{1,2,2,3}}      {{1,4},{1,2,2,3}}
  {{2},{1,2},{1,3}}  {{2},{1,1,2,3,3}}      {{2},{1,1,2,3,4}}
                     {{2,3},{1,1,2,3}}      {{2,3},{1,1,2,4}}
                     {{3},{1,1,2,2,3}}      {{2,4},{1,1,2,3}}
                     {{1},{1,2},{2,3,3}}    {{1},{1,2},{2,3,4}}
                     {{1},{1,3},{2,2,3}}    {{1},{2},{1,2,3,4}}
                     {{1},{2},{1,2,3,3}}    {{1,2},{1,3},{2,4}}
                     {{1,2},{1,3},{2,3}}    {{1,2},{1,4},{2,3}}
                     {{1},{2,3},{1,2,3}}    {{1},{2,3},{1,2,4}}
                     {{1},{3},{1,2,2,3}}    {{1},{2,4},{1,2,3}}
                     {{2},{1,2},{1,3,3}}    {{2},{1,2},{1,3,4}}
                     {{2},{1,3},{1,2,3}}    {{2},{1,3},{1,2,4}}
                     {{2},{2,3},{1,1,3}}    {{2},{1,4},{1,2,3}}
                     {{2},{3},{1,1,2,3}}
                     {{3},{1,2},{1,2,3}}
                     {{3},{1,3},{1,2,2}}
                     {{3},{2,3},{1,1,2}}
                     {{1},{2},{1,3},{2,3}}
                     {{1},{2},{3},{1,2,3}}
                     {{1},{3},{1,2},{2,3}}
                     {{2},{3},{1,2},{1,3}}

Crossrefs

Cf. A007718, A007719, A056156, A181821, A191970, A300913, A305193, A305936, A318284, A318286, A319557, A321272.

Sequence in context: A299200 A332819 A321272 * A030067 A181363 A105800

Adjacent sequences: A321267 A321268 A321269 * A321271 A321272 A321273

Keyword

nonn,more

Author

Gus Wiseman, Nov 01 2018

Status

approved