

A258026


Numbers k such that prime(k+2)  2*prime(k+1) + prime(k) < 0.


17



4, 6, 9, 11, 12, 16, 18, 19, 21, 24, 25, 27, 30, 32, 34, 37, 40, 42, 44, 47, 48, 51, 53, 56, 58, 59, 62, 63, 66, 68, 72, 74, 77, 80, 82, 84, 87, 88, 91, 92, 94, 97, 99, 101, 103, 106, 108, 111, 112, 114, 115, 119, 121, 125, 127, 128, 130, 132, 133, 135, 137
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OFFSET

1,1


COMMENTS

Positions of strict descents in the sequence of differences between primes. Partial sums of A333215.  Gus Wiseman, Mar 24 2020


LINKS

Clark Kimberling, Table of n, a(n) for n = 1..1000
Wikipedia, Longest increasing subsequence


EXAMPLE

The prime gaps split into the following maximal 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), ... Then a(n) is the nth partial sum of the lengths of these subsequences.  Gus Wiseman, Mar 24 2020


MATHEMATICA

u = Table[Sign[Prime[n+2]  2 Prime[n+1] + Prime[n]], {n, 1, 200}];
Flatten[Position[u, 0]] (* A064113 *)
Flatten[Position[u, 1]] (* A258025 *)
Flatten[Position[u, 1]] (* A258026 *)
Accumulate[Length/@Split[Differences[Array[Prime, 100]], LessEqual]]//Most (* Gus Wiseman, Mar 24 2020 *)


CROSSREFS

Partition of the positive integers: A064113, A258025, A258026;
Corresponding partition of the primes: A063535, A063535, A147812.
Adjacent terms differing by 1 correspond to strong prime quartets A054804.
The version for the Kolakoski sequence is A156242.
First differences are A333215 (if the first term is 0).
The version for strict ascents is A258025.
The version for weak ascents is A333230.
The version for weak descents is A333231.
Prime gaps are A001223.
Positions of adjacent equal prime gaps are A064113.
Weakly increasing runs of compositions in standard order are A124766.
Strictly decreasing runs of compositions in standard order are A124769.
Cf. A000040, A000720, A001221, A036263, A054819, A084758, A124765, A124768, A333212, A333213, A333214, A333256.
Sequence in context: A302990 A209920 A258979 * A246779 A160531 A190348
Adjacent sequences: A258023 A258024 A258025 * A258027 A258028 A258029


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Jun 05 2015


STATUS

approved



