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A318361
Number of strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.
18
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
1,4
LINKS
FORMULA
a(n) = A050326(A181821(n)).
a(prime(n)^k) = A188445(n, k). - Andrew Howroyd, Dec 17 2018
EXAMPLE
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}}
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}]
PROG
(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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 24 2018
STATUS
approved