A325759 Number of distinct frequencies in the frequency span of n.

0, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 3, 2, 1, 2, 1, 3, 3, 3, 1, 3, 2, 3, 2, 3, 1, 3, 1, 2, 3, 3, 4, 2, 1, 3, 3, 3, 1, 4, 1, 3, 3, 3, 1, 3, 2, 3, 3, 3, 1, 3, 4, 4, 3, 3, 1, 3, 1, 3, 3, 2, 4, 4, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 4, 4, 1, 4, 2, 3, 1, 3, 4, 3, 3

Offset: 1

Author

Gus Wiseman, May 19 2019

Keywords

• nonn

Comments

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 lengths of A325758.

Number of distinct entries in row n of A325757.

Cf. A001221, A001222, A056239, A071625, A112798, A181819, A182857, A323014, A325249, A325277, A325760.

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[Length[Union[freqspan[primeMS[n]]]],{n,100}]