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A335277 First index of strictly increasing prime quartets.

%I A335277 #6 May 30 2020 19:14:43
%S A335277 7,13,22,28,49,60,64,69,70,75,78,85,89,95,104,116,122,123,144,148,152,
%T A335277 155,173,178,182,195,201,206,212,215,219,225,226,230,236,237,244,253,
%U A335277 256,257,265,288,302,307,315,325,328,329,332,333,336,348,355,361,373
%N A335277 First index of strictly increasing prime quartets.
%C A335277 Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k) < g(k + 1) < g(k + 2).
%F A335277 prime(a(n)) = A054819(n).
%e A335277 The first 10 strictly increasing prime quartets:
%e A335277    17  19  23  29
%e A335277    41  43  47  53
%e A335277    79  83  89  97
%e A335277   107 109 113 127
%e A335277   227 229 233 239
%e A335277   281 283 293 307
%e A335277   311 313 317 331
%e A335277   347 349 353 359
%e A335277   349 353 359 367
%e A335277   379 383 389 397
%e A335277 For example, 107 is the 28th prime, and the primes (107,109,113,127) have differences (2,4,14), which are strictly increasing, so 28 is in the sequence.
%t A335277 ReplaceList[Array[Prime,100],{___,x_,y_,z_,t_,___}/;y-x<z-y<t-z:>PrimePi[x]]
%Y A335277 Prime gaps are A001223.
%Y A335277 Second prime gaps are A036263.
%Y A335277 Strictly decreasing prime quartets are A335278.
%Y A335277 Equal prime quartets are A090832.
%Y A335277 Weakly increasing prime quartets are A333383.
%Y A335277 Weakly decreasing prime quartets are A333488.
%Y A335277 Unequal prime quartets are A333490.
%Y A335277 Partially unequal prime quartets are A333491.
%Y A335277 Positions of adjacent equal prime gaps are A064113.
%Y A335277 Positions of strict ascents in prime gaps are A258025.
%Y A335277 Positions of strict descents in prime gaps are A258026.
%Y A335277 Positions of adjacent unequal prime gaps are A333214.
%Y A335277 Positions of weak ascents in prime gaps are A333230.
%Y A335277 Positions of weak descents in prime gaps are A333231.
%Y A335277 Lengths of maximal weakly decreasing sequences of prime gaps are A333212.
%Y A335277 Lengths of maximal strictly increasing sequences of prime gaps are A333253.
%Y A335277 Cf. A000040, A006560, A031217, A054800, A054819, A059044, A084758, A089180.
%K A335277 nonn
%O A335277 1,1
%A A335277 _Gus Wiseman_, May 30 2020