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

Author

Gus Wiseman, Aug 04 2024

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

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.

Programs

Mathematica

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

Formula

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