A325757 Irregular triangle read by rows giving the frequency span of n.

1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 4, 1, 1, 1, 3, 2, 2, 2, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 2, 2, 2, 6, 1, 1, 1, 2, 4, 1, 1, 2, 2, 3, 1, 1, 1, 1, 4, 7, 1, 1, 1, 1, 2, 2, 2, 2, 8, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 2, 2, 4, 1, 1, 1, 2, 5, 9, 1, 1, 1, 1, 1, 1, 2, 2, 3

Offset: 1

Author

Gus Wiseman, May 19 2019

Keywords

• nonn
• tabf

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).

Examples

			Triangle begins:
   1:
   2: 1
   3: 2
   4: 1 1 2
   5: 3
   6: 1 1 1 2 2
   7: 4
   8: 1 1 1 3
   9: 2 2 2
  10: 1 1 1 2 3
  11: 5
  12: 1 1 1 1 1 2 2 2
  13: 6
  14: 1 1 1 2 4
  15: 1 1 2 2 3
  16: 1 1 1 1 4
  17: 7
  18: 1 1 1 1 2 2 2 2
  19: 8
  20: 1 1 1 1 1 2 2 3
  21: 1 1 2 2 4
  22: 1 1 1 2 5
  23: 9
  24: 1 1 1 1 1 1 2 2 3
  25: 2 3 3
  26: 1 1 1 2 6
  27: 2 2 2 3
  28: 1 1 1 1 1 2 2 4
		

Crossrefs

Row lengths are A325249.

Run-lengths are A325758.

Number of distinct terms in row n is A325759(n).

Cf. A001221, A001222, A056239, A071625, A112798, A181819, A182857, A290822, A323014, A324843, A325277, A325755, 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[freqspan[primeMS[n]],{n,15}]