

A330666


Number of nonisomorphic balanced reduced multisystems whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.


5



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
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OFFSET

1,5


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}.


LINKS

Table of n, a(n) for n=1..30.


FORMULA

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


EXAMPLE

Nonisomorphic 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}}
Nonisomorphic 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 maximumdepth 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.
Sequence in context: A073546 A216975 A275666 * A319432 A115033 A214630
Adjacent sequences: A330663 A330664 A330665 * A330667 A330668 A330669


KEYWORD

nonn,more


AUTHOR

Gus Wiseman, Dec 30 2019


STATUS

approved



