A330664 Number of non-isomorphic balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.

1, 1, 1, 1, 1, 2, 2, 1, 4, 5, 5, 7, 16, 16, 27, 2, 61, 33, 272, 27, 123, 61, 1385, 27, 78, 272, 95, 123, 7936, 362

Offset

1,6

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

For n > 1, a(2^n) = a(prime(n)) = A000111(n - 1).

Example

Non-isomorphic representatives of the a(n) multisystems for n = 2, 3, 6, 9, 10, 12 (commas and outer brackets elided):
  1  11  {1}{12}  {{1}}{{1}{22}}  {{1}}{{1}{12}}  {{1}}{{1}{23}}
         {2}{11}  {{11}}{{2}{2}}  {{11}}{{1}{2}}  {{11}}{{2}{3}}
                  {{1}}{{2}{12}}  {{1}}{{2}{11}}  {{1}}{{2}{13}}
                  {{12}}{{1}{2}}  {{12}}{{1}{1}}  {{12}}{{1}{3}}
                                  {{2}}{{1}{11}}  {{2}}{{1}{13}}
                                                  {{2}}{{3}{11}}
                                                  {{23}}{{1}{1}}

Crossrefs

The non-maximal version is A330666.

The case of constant or strict atoms is A000111.

Labeled versions are A330728, A330665 (prime indices), and A330675 (strongly normal).

Non-isomorphic multiset partitions whose degrees are the prime indices of n are A318285.

Cf. A004114, A005121, A007716, A048816, A141268, A306186, A318846, A318848, A330470, A330474, A330663.

Sequence in context: A113547 A218580 A259697 * A330843 A115313 A048942

Adjacent sequences: A330661 A330662 A330663 * A330665 A330666 A330667

Keyword

nonn,more

Author

Gus Wiseman, Dec 28 2019

Status

approved