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A318284
Number of multiset partitions of a multiset whose multiplicities are the prime indices of n.
56
1, 1, 2, 2, 3, 4, 5, 5, 9, 7, 7, 11, 11, 12, 16, 15, 15, 26, 22, 21, 29, 19, 30, 36, 31, 30, 66, 38, 42, 52, 56, 52, 47, 45, 57, 92, 77, 67, 77, 74, 101, 98, 135, 64, 137, 97, 176, 135, 109, 109, 118, 105, 231, 249, 97, 141, 181, 139, 297, 198, 385, 195, 269
OFFSET
1,3
LINKS
FORMULA
a(n) = A001055(A181821(n)).
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 23 2018
STATUS
approved