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A333383 First index of weakly increasing prime quartets.

%I A333383 #6 May 16 2020 14:28:21
%S A333383 1,2,7,13,14,22,28,35,38,45,49,54,60,64,69,70,75,78,85,89,95,104,109,
%T A333383 116,117,122,123,144,148,152,155,159,160,163,164,173,178,182,183,184,
%U A333383 187,194,195,198,201,206,212,215,218,219,225,226,230,236,237,238,244
%N A333383 First index of weakly increasing prime quartets.
%C A333383 Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k) <= g(k + 1) <= g(k + 2).
%e A333383 The first 10 weakly increasing prime quartets:
%e A333383     2   3   5   7
%e A333383     3   5   7  11
%e A333383    17  19  23  29
%e A333383    41  43  47  53
%e A333383    43  47  53  59
%e A333383    79  83  89  97
%e A333383   107 109 113 127
%e A333383   149 151 157 163
%e A333383   163 167 173 179
%e A333383   197 199 211 223
%e A333383 For example, 43 is the 14th prime, and the primes (43,47,53,59) have differences (4,6,6), which are weakly increasing, so 14 is in the sequence.
%t A333383 ReplaceList[Array[Prime,100],{___,x_,y_,z_,t_,___}/;y-x<=z-y<=t-z:>PrimePi[x]]
%Y A333383 Prime gaps are A001223.
%Y A333383 Second prime gaps are A036263.
%Y A333383 Strictly decreasing prime quartets are A054804.
%Y A333383 Strictly increasing prime quartets are A054819.
%Y A333383 Equal prime quartets are A090832.
%Y A333383 Weakly increasing prime quartets are A333383 (this sequence).
%Y A333383 Weakly decreasing prime quartets are A333488.
%Y A333383 Unequal prime quartets are A333490.
%Y A333383 Partially unequal prime quartets are A333491.
%Y A333383 Positions of adjacent equal prime gaps are A064113.
%Y A333383 Positions of strict ascents in prime gaps are A258025.
%Y A333383 Positions of strict descents in prime gaps are A258026.
%Y A333383 Positions of adjacent unequal prime gaps are A333214.
%Y A333383 Positions of weak ascents in prime gaps are A333230.
%Y A333383 Positions of weak descents in prime gaps are A333231.
%Y A333383 Indices of weakly increasing rows of A066099 are A225620.
%Y A333383 Lengths of maximal weakly increasing subsequences of prime gaps: A333215.
%Y A333383 Lengths of maximal strictly decreasing subsequences of prime gaps: A333252.
%Y A333383 Cf. A000040, A006560, A031217, A054800, A059044, A084758, A089180, A333253.
%K A333383 nonn
%O A333383 1,2
%A A333383 _Gus Wiseman_, May 14 2020