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

1,2

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.

Links

Table of n, a(n) for n=1..84.

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

Example

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

Mathematica

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

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.

Sequence in context: A320250 A089141 A336084 * A245717 A248008 A327981

Adjacent sequences: A375125 A375126 A375127 * A375129 A375130 A375131

Keyword

nonn,tabf

Author

Gus Wiseman, Aug 04 2024

Status

approved