A375136 Number of maximal strictly increasing runs in the weakly increasing prime factors of n.

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset

1,4

Comments

For n > 1, this is one more than the number of adjacent equal terms in the multiset of prime factors of n.

Links

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

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

Formula

For n > 1, a(n) = A046660(n) + 1 = A001222(n) - A001221(n) + 1.

Example

The prime factors of 540 are {2,2,3,3,3,5}, with maximal strictly increasing runs ({2},{2,3},{3},{3,5}), so a(540) = 4.

Mathematica

Table[Length[Split[Flatten[ConstantArray@@@FactorInteger[n]], Less]], {n, 100}]

Crossrefs

For compositions we have A124768, row-lengths of A374683, sum A374684.

For sum of prime indices we have A374706.

Row-lengths of A375128.

A112798 lists prime indices:

- distinct A001221

- length A001222

- leader A055396

- sum A056239

- reverse A296150

Cf. A034296, A046660, A066328, A141199, A141809, A279790, A320324, A333213, A358836, A374700.

Sequence in context: A157754 A072411 A290107 * A212180 A091050 A005361

Adjacent sequences: A375131 A375132 A375135 * A375142 A375143 A375144

Keyword

nonn,new

Author

Gus Wiseman, Aug 04 2024

Status

approved