A330098 Number of distinct multisets of multisets that can be obtained by permuting the vertices of the multiset of multisets with MM-number n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Author

Gus Wiseman, Dec 09 2019

Keywords

• nonn

Comments

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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

a(n) is a divisor of A303975(n)!.

Examples

			The vertex-permutations of {{1,2},{2,3,3}} are:
  {{1,2},{1,3,3}}
  {{1,2},{2,3,3}}
  {{1,3},{1,2,2}}
  {{1,3},{2,2,3}}
  {{2,3},{1,1,2}}
  {{2,3},{1,1,3}}
so a(4927) = 6.
		

Crossrefs

Positions of 1's are A330232.

Positions of first appearances are A330230 and A330233.

The BII-number version is A330231.

Cf. A001055, A003238, A007716, A055621, A056239, A112798, A302242, A303975, A322847, A330194, A330218, A330223, A330227, A330236.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[graprms[primeMS/@primeMS[n]]],{n,100}]