A325758 Irregular triangle read by rows giving the frequency span signature of n.

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, 1, 5, 1, 2, 2, 1, 3, 1, 1, 2, 1, 1, 1, 2, 5, 1, 3, 1, 1, 2, 2, 1, 6, 1, 2, 1, 4, 1, 1, 1

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).

Links

Table of n, a(n) for n=1..87.

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

Row sums are A325249.

Row lengths are A325759.

Run-lengths of A325757.

Row n is the unsorted prime signature of A325760(n).

Cf. A001221, A001222, A056239, A071625, A112798, A181819, A182857, A290822, A323014,A325277.

Sequence in context: A235682 A324830 A141110 * A280073 A327523 A190770

Adjacent sequences: A325755 A325756 A325757 * A325759 A325760 A325761

Keyword

nonn,tabf

Author

Gus Wiseman, May 19 2019

Status

approved