%I #5 May 16 2020 14:28:42

%S 7,8,10,11,13,17,18,19,20,22,23,24,28,30,31,32,34,40,42,44,47,49,50,

%T 51,52,57,58,59,60,61,62,64,65,66,67,68,69,70,75,76,78,79,82,83,85,86,

%U 87,89,90,91,94,95,96,97,98,99,104,111,112,113,114,115,116,119

%N First index of unequal prime quartets.

%C Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k), g(k + 1), and g(k + 2) are all different.

%e The first 10 unequal prime quartets:

%e 17 19 23 29

%e 19 23 29 31

%e 29 31 37 41

%e 31 37 41 43

%e 41 43 47 53

%e 59 61 67 71

%e 61 67 71 73

%e 67 71 73 79

%e 71 73 79 83

%e 79 83 89 97

%e For example, 83 is the 23rd prime, and the primes (83,89,97,101) have differences (6,8,4), which are all distinct, so 23 is in the sequence.

%t ReplaceList[Array[Prime,100],{___,x_,y_,z_,t_,___}/;y-x!=z-y!=t-z:>PrimePi[x]]

%Y Primes are A000040.

%Y Prime gaps are A001223.

%Y Second prime gaps are A036263.

%Y Indices of unequal rows of A066099 are A233564.

%Y Lengths of maximal anti-run subsequences of prime gaps are A333216.

%Y Lengths of maximal runs of prime gaps are A333254.

%Y Maximal anti-runs in standard compositions are counted by A333381.

%Y Indices of anti-run rows of A066099 are A333489.

%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.

%Y Unequal prime quartets are A333490 (this sequence).

%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 Cf. A006560, A031217, A054800, A059044, A084758, A089180, A124767, A333215.

%K nonn

%O 1,1

%A _Gus Wiseman_, May 15 2020