OFFSET
1,4
COMMENTS
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also the number of size-preserving permutations of set multipartitions (multisets of sets) of a multiset (such as row n of A305936) whose multiplicities are the prime indices of n.
EXAMPLE
The sum of coefficients of e(211) = 2m(22) + m(31) + 5m(211) + 12m(1111) is a(12) = 20.
The a(2) = 1 through a(9) = 9 size-preserving permutations of set multipartitions:
{1} {1}{1} {12} {1}{1}{1} {1}{12} {1}{1}{1}{1} {123} {12}{12}
{1}{2} {1}{1}{2} {1}{23} {1}{2}{12}
{2}{1} {1}{2}{1} {2}{13} {2}{1}{12}
{2}{1}{1} {3}{12} {1}{1}{2}{2}
{1}{2}{3} {1}{2}{1}{2}
{1}{3}{2} {1}{2}{2}{1}
{2}{1}{3} {2}{1}{1}{2}
{2}{3}{1} {2}{1}{2}{1}
{3}{1}{2} {2}{2}{1}{1}
{3}{2}{1}
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Table[Sum[Times@@Factorial/@Length/@Split[Sort[Length/@mtn, Greater]]/Times@@Factorial/@Length/@Split[mtn], {mtn, Select[mps[nrmptn[n]], And@@UnsameQ@@@#&]}], {n, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 19 2018
STATUS
approved