A318361 Number of strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.

1, 1, 0, 2, 0, 1, 0, 5, 1, 0, 0, 4, 0, 0, 0, 15, 0, 5, 0, 1, 0, 0, 0, 16, 0, 0, 8, 0, 0, 2, 0, 52, 0, 0, 0, 23, 0, 0, 0, 7, 0, 0, 0, 0, 5, 0, 0, 68, 0, 1, 0, 0, 0, 40, 0, 1, 0, 0, 0, 14, 0, 0, 1, 203, 0, 0, 0, 0, 0, 0, 0, 111, 0, 0, 4, 0, 0, 0, 0, 41, 80, 0, 0

Offset: 1

Author

Gus Wiseman, Aug 24 2018

Keywords

• nonn

Examples

			The a(24) = 16 sets of sets with multiset union {1,1,2,3,4}:
  {{1},{1,2,3,4}}
  {{1,2},{1,3,4}}
  {{1,3},{1,2,4}}
  {{1,4},{1,2,3}}
  {{1},{2},{1,3,4}}
  {{1},{3},{1,2,4}}
  {{1},{4},{1,2,3}}
  {{1},{1,2},{3,4}}
  {{1},{1,3},{2,4}}
  {{1},{1,4},{2,3}}
  {{2},{1,3},{1,4}}
  {{3},{1,2},{1,4}}
  {{4},{1,2},{1,3}}
  {{1},{2},{3},{1,4}}
  {{1},{2},{4},{1,3}}
  {{1},{3},{4},{1,2}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A007716, A045778, A049311, A050326, A116540, A292432, A292444, A293511.

Cf. A318283, A318284, A318286, A318287, A318360, A318361, A318370, A318371.

Programs

Mathematica

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[sqfacs[Times@@Prime/@nrmptn[n]]],{n,90}]

PARI

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
sig(n)={my(f=factor(n)); concat(vector(#f~, i, vector(f[i, 2], j, primepi(f[i, 1]))))}
count(sig)={my(r=0, A=O(x*x^vecmax(sig))); for(n=1, vecsum(sig)+1, my(s=0); forpart(p=n, my(q=prod(i=1, #p, 1 + x^p[i] + A)); s+=prod(i=1, #sig, polcoef(q, sig[i]))*(-1)^#p*permcount(p)); r+=(-1)^n*s/n!); r/2}
a(n)={if(n==1, 1, my(s=sig(n)); if(#s==1, s[1]==1, count(sig(n))))} \\ Andrew Howroyd, Dec 18 2018

Formula

a(n) = A050326(A181821(n)).

a(prime(n)^k) = A188445(n, k). - Andrew Howroyd, Dec 17 2018