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First index of strictly decreasing prime quartets.
3

%I #5 May 30 2020 19:14:48

%S 11,18,24,47,58,62,87,91,111,114,127,132,146,150,157,180,210,223,228,

%T 232,242,259,260,263,269,274,275,282,283,284,299,300,309,321,344,350,

%U 351,363,364,367,368,369,375,378,382,388,393,399,406,409,413,431,442,446

%N First index of strictly decreasing 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)) = A054804(n).

%e The first 10 strictly decreasing prime quartets:

%e 31 37 41 43

%e 61 67 71 73

%e 89 97 101 103

%e 211 223 227 229

%e 271 277 281 283

%e 293 307 311 313

%e 449 457 461 463

%e 467 479 487 491

%e 607 613 617 619

%e 619 631 641 643

%e For example, 211 is the 47th prime, and the primes (211,223,227,229) have differences (12,4,2), which are strictly decreasing, so 47 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 increasing prime quartets are A335277.

%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 Indices of strictly decreasing rows of A066099 are A333256.

%Y Lengths of maximal weakly increasing sequences of prime gaps are A333215.

%Y Lengths of maximal strictly decreasing sequences of prime gaps are A333252.

%Y Cf. A000040, A006560, A031217, A054800, A054804, A059044, A084758, A089180, A333253.

%K nonn

%O 1,1

%A _Gus Wiseman_, May 30 2020