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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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..700 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 CROSSREFS Cf. A001055, A001970, A007716, A181821, A219727, A255906, A269134, A305936. Cf. A317532, A318283, A318285, A318286, A318360, A318361. Sequence in context: A179255 A332285 A324325 * A241321 A093936 A329904 Adjacent sequences:  A318281 A318282 A318283 * A318285 A318286 A318287 KEYWORD nonn AUTHOR Gus Wiseman, Aug 23 2018 STATUS approved

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Last modified April 21 07:26 EDT 2021. Contains 343146 sequences. (Running on oeis4.)