A325760 Heinz number of the frequency span of n.

1, 2, 3, 12, 5, 72, 7, 40, 27, 120, 11, 864, 13, 168, 180, 112, 17, 1296, 19, 1440, 252, 264, 23, 2880, 75, 312, 135, 2016, 29, 1200, 31, 352, 396, 408, 420, 972, 37, 456, 468, 4800, 41, 1680, 43, 3168, 3240, 552, 47, 8064, 147, 3600, 612, 3744, 53, 6480, 660

Offset: 1

Author

Gus Wiseman, May 19 2019

Keywords

• nonn

Comments

The Heinz number of a positive integer sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

We define the frequency span of an integer partition to be the partition itself if it has no or only one block, and otherwise it is the multiset union of the partition and the frequency span of its multiplicities. For example, the frequency span of (3,2,2,1) is {1,2,2,3} U {1,1,2} U {1,2} U {1,1} U {2} = {1,1,1,1,1,1,2,2,2,2,2,3}. The frequency span of a positive integer n is the frequency span of its prime indices (row n of A296150).

Crossrefs

Row-products of A325277.

The prime indices of a(n) are row n of A325757.

The unsorted prime signature of a(n) is row n of A325758.

Cf. A001221, A001222, A056239, A071625, A112798, A181819, A182857, A323014, A324843, A325248, A325249, A325759.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
freqspan[ptn_]:=If[Length[ptn]<=1,ptn,Sort[Join[ptn,freqspan[Sort[Length/@Split[ptn]]]]]];
Table[Times@@Prime/@freqspan[primeMS[n]],{n,30}]