OFFSET
1,3
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 is the frequency span of its prime indices (row n of A296150). Row n of this triangle gives an unsorted list of the multiplicities in the frequency span of n. For example, the frequency span of 30 is {1,1,1,1,2,3,3}, so row 30 is (4,1,2).
EXAMPLE
Triangle begins:
1
1
2 1
1
3 2
1
3 1
3
3 1 1
1
5 3
1
3 1 1
2 2 1
4 1
1
4 4
1
5 2 1
2 2 1
3 1 1
1
6 2 1
1 2
3 1 1
3 1
5 2 1
1
4 1 2
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/@Split[freqspan[primeMS[n]]], {n, 30}]
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, May 19 2019
STATUS
approved