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A333488 First index of weakly decreasing prime quartets.

%I A333488 #7 May 16 2020 14:28:35
%S A333488 11,15,18,24,36,39,46,47,53,54,55,58,62,72,73,87,91,101,102,106,107,
%T A333488 110,111,114,118,127,128,129,132,146,150,157,180,186,193,199,210,217,
%U A333488 223,228,232,239,242,259,260,263,269,270,271,274,275,282,283,284,290
%N A333488 First index of weakly decreasing prime quartets.
%C A333488 Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k) >= g(k + 1) >= g(k + 2).
%e A333488 The first 10 weakly decreasing prime quartets:
%e A333488    31  37  41  43
%e A333488    47  53  59  61
%e A333488    61  67  71  73
%e A333488    89  97 101 103
%e A333488   151 157 163 167
%e A333488   167 173 179 181
%e A333488   199 211 223 227
%e A333488   211 223 227 229
%e A333488   241 251 257 263
%e A333488   251 257 263 269
%e A333488 For example, 241 is the 53rd prime, and the primes (241,251,257,263) have differences (10,6,6), which are weakly decreasing, so 53 is in the sequence.
%t A333488 ReplaceList[Array[Prime,100],{___,x_,y_,z_,t_,___}/;y-x>=z-y>=t-z:>PrimePi[x]]
%Y A333488 Prime gaps are A001223.
%Y A333488 Second prime gaps are A036263.
%Y A333488 Strictly decreasing prime quartets are A054804.
%Y A333488 Strictly increasing prime quartets are A054819.
%Y A333488 Equal prime quartets are A090832.
%Y A333488 Weakly increasing prime quartets are A333383.
%Y A333488 Weakly decreasing prime quartets are A333488 (this sequence).
%Y A333488 Unequal prime quartets are A333490.
%Y A333488 Partially unequal prime quartets are A333491.
%Y A333488 Positions of adjacent equal prime gaps are A064113.
%Y A333488 Positions of strict ascents in prime gaps are A258025.
%Y A333488 Positions of strict descents in prime gaps are A258026.
%Y A333488 Positions of adjacent unequal prime gaps are A333214.
%Y A333488 Positions of weak ascents in prime gaps are A333230.
%Y A333488 Positions of weak descents in prime gaps are A333231.
%Y A333488 Indices of weakly decreasing rows of A066099 are A114994.
%Y A333488 Lengths of maximal weakly decreasing subsequences of prime gaps: A333212.
%Y A333488 Lengths of maximal strictly increasing subsequences of prime gaps: A333253.
%Y A333488 Cf. A000040, A006560, A031217, A054800, A059044, A084758, A089180.
%K A333488 nonn
%O A333488 1,1
%A A333488 _Gus Wiseman_, May 15 2020