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