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A333491 First index of partially unequal prime quartets.

%I A333491 #6 Aug 05 2025 17:42:58
%S A333491 3,4,5,6,7,8,9,10,11,12,13,16,17,18,19,20,21,22,23,24,25,26,27,28,29,
%T A333491 30,31,32,33,34,37,40,41,42,43,44,47,48,49,50,51,52,56,57,58,59,60,61,
%U A333491 62,63,64,65,66,67,68,69,70,71,74,75,76,77,78,79,80,81,82
%N A333491 First index of partially unequal prime quartets.
%C A333491 Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k) != g(k + 1) != g(k + 2), but we may have g(k) = g(k + 2).
%e A333491 The first 10 partially unequal prime quartets:
%e A333491    5  7 11 13
%e A333491    7 11 13 17
%e A333491   11 13 17 19
%e A333491   13 17 19 23
%e A333491   17 19 23 29
%e A333491   19 23 29 31
%e A333491   23 29 31 37
%e A333491   29 31 37 41
%e A333491   31 37 41 43
%e A333491   37 41 43 47
%t A333491 ReplaceList[Array[Prime,100],{___,x_,y_,z_,t_,___}/;y-x!=z-y&&z-y!=t-z:>PrimePi[x]]
%t A333491 PrimePi[#]&/@(Select[Partition[Prime[Range[90]],4,1],#[[2]]-#[[1]]!=#[[3]]-#[[2]]&&#[[3]]-#[[2]]!=#[[4]]-#[[3]]&][[;;,1]]) (* _Harvey P. Dale_, Aug 05 2025 *)
%Y A333491 Primes are A000040.
%Y A333491 Prime gaps are A001223.
%Y A333491 Second prime gaps are A036263.
%Y A333491 Indices of unequal rows of A066099 are A233564.
%Y A333491 Lengths of maximal anti-runs of prime gaps are A333216.
%Y A333491 Lengths of maximal runs of prime gaps are A333254.
%Y A333491 Maximal anti-runs in standard compositions are counted by A333381.
%Y A333491 Indices of anti-run rows of A066099 are A333489.
%Y A333491 Strictly decreasing prime quartets are A054804.
%Y A333491 Strictly increasing prime quartets are A054819.
%Y A333491 Equal prime quartets are A090832.
%Y A333491 Weakly increasing prime quartets are A333383.
%Y A333491 Weakly decreasing prime quartets are A333488.
%Y A333491 Unequal prime quartets are A333490.
%Y A333491 Partially unequal prime quartets are A333491 (this sequence).
%Y A333491 Positions of adjacent equal prime gaps are A064113.
%Y A333491 Positions of strict ascents in prime gaps are A258025.
%Y A333491 Positions of strict descents in prime gaps are A258026.
%Y A333491 Positions of adjacent unequal prime gaps are A333214.
%Y A333491 Positions of weak ascents in prime gaps are A333230.
%Y A333491 Positions of weak descents in prime gaps are A333231.
%Y A333491 Cf. A006560, A031217, A054800, A059044, A084758, A089180, A124767, A333215.
%K A333491 nonn
%O A333491 1,1
%A A333491 _Gus Wiseman_, May 15 2020