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A318360
Number of set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.
31
1, 1, 1, 2, 1, 2, 1, 5, 3, 2, 1, 6, 1, 2, 3, 15, 1, 9, 1, 6, 3, 2, 1, 21, 4, 2, 16, 6, 1, 10, 1, 52, 3, 2, 4, 35, 1, 2, 3, 22, 1, 10, 1, 6, 19, 2, 1, 83, 5, 13, 3, 6, 1, 66, 4, 22, 3, 2, 1, 41, 1, 2, 20, 203, 4, 10, 1, 6, 3, 14, 1, 153, 1, 2, 26, 6, 5, 10, 1
OFFSET
1,4
LINKS
FORMULA
a(n) = A050320(A181821(n)).
From Andrew Howroyd, Dec 10 2018:(Start)
a(p) = 1 for prime(p).
a(prime(i)*prime(j)) = min(i,j) + 1.
a(prime(n)^k) = A188392(n,k). (End)
EXAMPLE
The a(12) = 6 set multipartitions of {1,1,2,3}:
{{1},{1,2,3}}
{{1,2},{1,3}}
{{1},{1},{2,3}}
{{1},{2},{1,3}}
{{1},{3},{1,2}}
{{1},{1},{2},{3}}
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, 80}]
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(n=vecsum(sig), s=0); forpart(p=n, my(q=prod(i=1, #p, 1 + x^p[i] + O(x*x^n))); s+=prod(i=1, #sig, polcoef(q, sig[i]))*permcount(p)); s/n!}
a(n)={if(n==1, 1, my(s=sig(n)); if(#s<=2, if(#s==1, 1, min(s[1], s[2])+1), count(sig(n))))} \\ Andrew Howroyd, Dec 10 2018
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 24 2018
STATUS
approved