A375128 Irregular triangle read by rows where row n lists the minima of maximal strictly increasing runs in the weakly increasing prime indices of n.

1, 2, 1, 1, 3, 1, 4, 1, 1, 1, 2, 2, 1, 5, 1, 1, 6, 1, 2, 1, 1, 1, 1, 7, 1, 2, 8, 1, 1, 2, 1, 9, 1, 1, 1, 3, 3, 1, 2, 2, 2, 1, 1, 10, 1, 11, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 12, 1, 2, 1, 1, 1, 13, 1, 14, 1, 1, 2, 2, 1, 15, 1, 1, 1, 1, 4, 4, 1, 3, 2, 1, 1, 16

Offset: 1

Author

Gus Wiseman, Aug 04 2024

Keywords

• nonn
• tabf

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The minima of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.

Examples

			The prime indices of 540 are {1,1,2,2,2,3}, with strictly increasing runs ({1},{1,2},{2},{2,3}), with minima (1,1,2,2), which is row 540.
Triangle begins:
   1:
   2:  1
   3:  2
   4:  1  1
   5:  3
   6:  1
   7:  4
   8:  1  1  1
   9:  2  2
  10:  1
  11:  5
  12:  1  1
  13:  6
  14:  1
  15:  2
  16:  1  1  1  1
		

Links

• Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Crossrefs

Row-minima are A055396.

Row-sums are A374706.

Row-lengths are A375136.

For leaders of constant runs we have A304038, row-sums A066328.

For compositions we have A374683, row-sums of A374684 (length A124768).

A112798 lists prime indices:

- length A001222, distinct A001221

- leader A055396

- sum A056239

- reverse A296150

Cf. A034296, A141199, A218482, A279790, A320324, A333213, A358836, A374634, A374700, A374758, A375133.

Programs

Mathematica

Table[If[n==1,{},First/@Split[Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]],Less]],{n,100}]