A330098 - OEIS

%I #5 Dec 10 2019 12:15:35

%S 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,

%T 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,

%U 2,2,2,1,1,2,2,1,2,2,1,1,1,1,1,1,1,1,2

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

%C 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}}.

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

%e The vertex-permutations of {{1,2},{2,3,3}} are:

%e   {{1,2},{1,3,3}}

%e   {{1,2},{2,3,3}}

%e   {{1,3},{1,2,2}}

%e   {{1,3},{2,2,3}}

%e   {{2,3},{1,1,2}}

%e   {{2,3},{1,1,3}}

%e so a(4927) = 6.

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];

%t Table[Length[graprms[primeMS/@primeMS[n]]],{n,100}]

%Y Positions of 1's are A330232.

%Y Positions of first appearances are A330230 and A330233.

%Y The BII-number version is A330231.

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

%K nonn

%O 1,35

%A _Gus Wiseman_, Dec 09 2019