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First index of weakly decreasing prime quartets.
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%I #7 May 16 2020 14:28:35

%S 11,15,18,24,36,39,46,47,53,54,55,58,62,72,73,87,91,101,102,106,107,

%T 110,111,114,118,127,128,129,132,146,150,157,180,186,193,199,210,217,

%U 223,228,232,239,242,259,260,263,269,270,271,274,275,282,283,284,290

%N First index of weakly 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).

%e The first 10 weakly decreasing prime quartets:

%e 31 37 41 43

%e 47 53 59 61

%e 61 67 71 73

%e 89 97 101 103

%e 151 157 163 167

%e 167 173 179 181

%e 199 211 223 227

%e 211 223 227 229

%e 241 251 257 263

%e 251 257 263 269

%e 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 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.

%Y Weakly decreasing prime quartets are A333488 (this sequence).

%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 decreasing rows of A066099 are A114994.

%Y Lengths of maximal weakly decreasing subsequences of prime gaps: A333212.

%Y Lengths of maximal strictly increasing subsequences of prime gaps: A333253.

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

%K nonn

%O 1,1

%A _Gus Wiseman_, May 15 2020