OFFSET
1,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..700
FORMULA
a(prime(n)^k) = A219727(n,k). - Andrew Howroyd, Dec 10 2018
EXAMPLE
The a(12) = 11 multiset partitions of {1,1,2,3}:
{{1,1,2,3}}
{{1},{1,2,3}}
{{2},{1,1,3}}
{{3},{1,1,2}}
{{1,1},{2,3}}
{{1,2},{1,3}}
{{1},{1},{2,3}}
{{1},{2},{1,3}}
{{1},{3},{1,2}}
{{2},{3},{1,1}}
{{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}]]]]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[facs[Times@@Prime/@nrmptn[n]]], {n, 60}]
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), A=O(x*x^vecmax(sig)), s=0); forpart(p=n, my(q=1/prod(i=1, #p, 1 - x^p[i] + A)); 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==1, numbpart(s[1]), count(sig(n))))} \\ Andrew Howroyd, Dec 10 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 23 2018
STATUS
approved