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.
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
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 01 2018
STATUS
approved