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%I #20 Feb 28 2024 10:21:11
%S 4,6,9,11,12,16,18,19,21,24,25,27,30,32,34,37,40,42,44,47,48,51,53,56,
%T 58,59,62,63,66,68,72,74,77,80,82,84,87,88,91,92,94,97,99,101,103,106,
%U 108,111,112,114,115,119,121,125,127,128,130,132,133,135,137
%N Numbers k such that prime(k+2) - 2*prime(k+1) + prime(k) < 0.
%C Positions of strict descents in the sequence of differences between primes. Partial sums of A333215. - _Gus Wiseman_, Mar 24 2020
%H Clark Kimberling, <a href="/A258026/b258026.txt">Table of n, a(n) for n = 1..1000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Longest_increasing_subsequence">Longest increasing subsequence</a>
%e 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 n-th partial sum of the lengths of these subsequences. - _Gus Wiseman_, Mar 24 2020
%t u = Table[Sign[Prime[n+2] - 2 Prime[n+1] + Prime[n]], {n, 1, 200}];
%t Flatten[Position[u, 0]] (* A064113 *)
%t Flatten[Position[u, 1]] (* A258025 *)
%t Flatten[Position[u, -1]] (* A258026 *)
%t Accumulate[Length/@Split[Differences[Array[Prime,100]],LessEqual]]//Most (* _Gus Wiseman_, Mar 24 2020 *)
%o (Python)
%o from itertools import count, islice
%o from sympy import prime, nextprime
%o def A258026_gen(startvalue=1): # generator of terms >= startvalue
%o c = max(startvalue,1)
%o p = prime(c)
%o q = nextprime(p)
%o r = nextprime(q)
%o for k in count(c):
%o if p+r<(q<<1):
%o yield k
%o p, q, r = q, r, nextprime(r)
%o A258026_list = list(islice(A258026_gen(),20)) # _Chai Wah Wu_, Feb 27 2024
%Y Partition of the positive integers: A064113, A258025, A258026;
%Y Corresponding partition of the primes: A063535, A063535, A147812.
%Y Adjacent terms differing by 1 correspond to strong prime quartets A054804.
%Y The version for the Kolakoski sequence is A156242.
%Y First differences are A333215 (if the first term is 0).
%Y The version for strict ascents is A258025.
%Y The version for weak ascents is A333230.
%Y The version for weak descents is A333231.
%Y Prime gaps are A001223.
%Y Positions of adjacent equal prime gaps are A064113.
%Y Weakly increasing runs of compositions in standard order are A124766.
%Y Strictly decreasing runs of compositions in standard order are A124769.
%Y Cf. A000040, A000720, A001221, A036263, A054819, A084758, A124765, A124768, A333212, A333213, A333214, A333256.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, Jun 05 2015
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