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First index of strictly increasing prime quartets.
3

%I #6 May 30 2020 19:14:43

%S 7,13,22,28,49,60,64,69,70,75,78,85,89,95,104,116,122,123,144,148,152,

%T 155,173,178,182,195,201,206,212,215,219,225,226,230,236,237,244,253,

%U 256,257,265,288,302,307,315,325,328,329,332,333,336,348,355,361,373

%N First index of strictly 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).

%F prime(a(n)) = A054819(n).

%e The first 10 strictly increasing prime quartets:

%e 17 19 23 29

%e 41 43 47 53

%e 79 83 89 97

%e 107 109 113 127

%e 227 229 233 239

%e 281 283 293 307

%e 311 313 317 331

%e 347 349 353 359

%e 349 353 359 367

%e 379 383 389 397

%e 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 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 A335278.

%Y Equal prime quartets are A090832.

%Y Weakly increasing prime quartets are A333383.

%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 Lengths of maximal weakly decreasing sequences of prime gaps are A333212.

%Y Lengths of maximal strictly increasing sequences of prime gaps are A333253.

%Y Cf. A000040, A006560, A031217, A054800, A054819, A059044, A084758, A089180.

%K nonn

%O 1,1

%A _Gus Wiseman_, May 30 2020