A321272 Number of connected multiset partitions with multiset density -1, of a multiset whose multiplicities are the prime indices of n.

0, 1, 2, 1, 3, 2, 5, 1, 4, 4, 7, 3, 11, 7, 8, 1, 15, 8, 22, 7, 14, 12, 30, 5, 16, 19, 20, 14, 42, 18, 56, 1, 24, 30, 28, 18, 77, 45, 38, 14

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}.

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

Links

Table of n, a(n) for n=1..40.

Formula

a(prime(n)) = A000041(n).

Example

Non-isomorphic representatives of the a(2) = 1 through a(15) = 8 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}}
           {{1}{122}}    {{1}{112}}    {{1}{1111}}
           {{2}{112}}    {{11}{12}}    {{11}{111}}
           {{1}{2}{12}}  {{1}{1}{12}}  {{1}{1}{111}}
                                       {{1}{11}{11}}
                                       {{1}{1}{1}{11}}
                                       {{1}{1}{1}{1}{1}}
.
  {{1123}}    {{111111}}            {{11112}}        {{11122}}
  {{1}{123}}  {{1}{11111}}          {{1}{1112}}      {{1}{1122}}
  {{12}{13}}  {{11}{1111}}          {{11}{112}}      {{11}{122}}
              {{111}{111}}          {{12}{111}}      {{2}{1112}}
              {{1}{1}{1111}}        {{1}{1}{112}}    {{1}{1}{122}}
              {{1}{11}{111}}        {{1}{11}{12}}    {{1}{2}{112}}
              {{11}{11}{11}}        {{1}{1}{1}{12}}  {{2}{11}{12}}
              {{1}{1}{1}{111}}                       {{1}{1}{2}{12}}
              {{1}{1}{11}{11}}
              {{1}{1}{1}{1}{11}}
              {{1}{1}{1}{1}{1}{1}}

Crossrefs

Cf. A007718, A181821, A303837, A304382, A305081, A305936, A318284, A321155, A321229, A321253, A321270, A321271.

Sequence in context: A250479 A299200 A332819 * A321270 A030067 A181363

Adjacent sequences: A321269 A321270 A321271 * A321273 A321274 A321275

Keyword

nonn,more

Author

Gus Wiseman, Nov 01 2018

Status

approved