%I #9 Mar 15 2020 22:25:30

%S 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,

%T 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,

%U 3,3,1,4,4,2,2,2,3,2,3,1,5,3,2,2,4,3,3

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

%C Prime gaps are differences between adjacent prime numbers.

%F 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)).

%e 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), ...

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

%Y Prime gaps are A001223.

%Y Ones correspond to strong prime quartets A054804.

%Y Weakly increasing runs of compositions in standard order are A124766.

%Y First differences of A258026 (with zero prepended).

%Y The version for the Kolakoski sequence is A332875.

%Y The weakly decreasing version is A333212.

%Y The unequal version is A333216.

%Y Positions of weak ascents in prime gaps are A333230.

%Y The strictly decreasing version is A333252.

%Y The strictly increasing version is A333253.

%Y The equal version is A333254.

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Mar 14 2020