A318286 - OEIS

A318286

Number of strict multiset partitions of a multiset whose multiplicities are the prime indices of n.

1, 1, 1, 2, 2, 3, 2, 5, 5, 5, 3, 9, 4, 7, 9, 15, 5, 18, 6, 16, 14, 10, 8, 31, 17, 14, 40, 25, 10, 34, 12, 52, 21, 19, 27, 70, 15, 25, 31, 59, 18, 57, 22, 38, 80, 33, 27, 120, 46, 67, 44, 56, 32, 172, 42, 100, 61, 43, 38, 141, 46, 55, 143, 203, 64, 91, 54, 80

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OFFSET

1,4

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..500

FORMULA

a(n) = A045778(A181821(n)).

a(prime(n)^k) = A219585(n, k). - Andrew Howroyd, Dec 17 2018

MATHEMATICA

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];

strfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[strfacs[n/d], Min@@#>d&]], {d, Rest[Divisors[n]]}]];

Table[Length[strfacs[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(r=0, A=O(x*x^vecmax(sig))); for(n=1, vecsum(sig)+1, my(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]))*(-1)^#p*permcount(p)); r+=(-1)^n*s/n!); r/2}

a(n)={if(n==1, 1, count(sig(n)))} \\ Andrew Howroyd, Dec 18 2018

CROSSREFS

Cf. A001055, A007716, A045778, A181821, A305936, A316980, A317776.

Cf. A318283, A318284, A318285, A318287, A318360, A318361.

Sequence in context: A090794 A254858 A050323 * A214646 A340693 A275382

Adjacent sequences: A318283 A318284 A318285 * A318287 A318288 A318289

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 23 2018

STATUS

approved