A333383 - OEIS

%I #6 May 16 2020 14:28:21

%S 1,2,7,13,14,22,28,35,38,45,49,54,60,64,69,70,75,78,85,89,95,104,109,

%T 116,117,122,123,144,148,152,155,159,160,163,164,173,178,182,183,184,

%U 187,194,195,198,201,206,212,215,218,219,225,226,230,236,237,238,244

%N First index of weakly 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).

%e The first 10 weakly increasing prime quartets:

%e     2   3   5   7

%e     3   5   7  11

%e    17  19  23  29

%e    41  43  47  53

%e    43  47  53  59

%e    79  83  89  97

%e   107 109 113 127

%e   149 151 157 163

%e   163 167 173 179

%e   197 199 211 223

%e For example, 43 is the 14th prime, and the primes (43,47,53,59) have differences (4,6,6), which are weakly increasing, so 14 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 (this sequence).

%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 weakly increasing rows of A066099 are A225620.

%Y Lengths of maximal weakly increasing subsequences of prime gaps: A333215.

%Y Lengths of maximal strictly decreasing subsequences of prime gaps: A333252.

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

%K nonn

%O 1,2

%A _Gus Wiseman_, May 14 2020