A333215 Lengths of maximal weakly increasing subsequences in the sequence of prime gaps (A001223).

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

Offset: 1

Author

Gus Wiseman, Mar 14 2020

Keywords

• nonn

Comments

Prime gaps are differences between adjacent prime numbers.

Examples

			The prime gaps split into the following weakly increasing subsequences: (1,2,2,4), (2,4), (2,4,6), (2,6), (4), (2,4,6,6), (2,6), (4), (2,6), (4,6,8), (4), (2,4), (2,4,14), ...
		

Crossrefs

Prime gaps are A001223.

Ones correspond to strong prime quartets A054804.

Weakly increasing runs of compositions in standard order are A124766.

First differences of A258026 (with zero prepended).

The version for the Kolakoski sequence is A332875.

The weakly decreasing version is A333212.

The unequal version is A333216.

Positions of weak ascents in prime gaps are A333230.

The strictly decreasing version is A333252.

The strictly increasing version is A333253.

The equal version is A333254.

Cf. A000040, A000720, A036263, A054819, A064113, A084758, A124765, A124768, A258025, A333213, A333214.

Programs

Mathematica

Length/@Split[Differences[Array[Prime,100]],#1<=#2&]//Most

Formula

Ones correspond to strong prime quartets (A054804), so the sum of terms up to but not including the n-th one is A000720(A054804(n - 1)).