A333490 First index of unequal prime quartets.

7, 8, 10, 11, 13, 17, 18, 19, 20, 22, 23, 24, 28, 30, 31, 32, 34, 40, 42, 44, 47, 49, 50, 51, 52, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 75, 76, 78, 79, 82, 83, 85, 86, 87, 89, 90, 91, 94, 95, 96, 97, 98, 99, 104, 111, 112, 113, 114, 115, 116, 119

Offset: 1

Author

Gus Wiseman, May 15 2020

Keywords

• nonn

Comments

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.

Examples

			The first 10 unequal prime quartets:
  17  19  23  29
  19  23  29  31
  29  31  37  41
  31  37  41  43
  41  43  47  53
  59  61  67  71
  61  67  71  73
  67  71  73  79
  71  73  79  83
  79  83  89  97
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.
		

Crossrefs

Primes are A000040.

Prime gaps are A001223.

Second prime gaps are A036263.

Indices of unequal rows of A066099 are A233564.

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

Lengths of maximal runs of prime gaps are A333254.

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

Indices of anti-run rows of A066099 are A333489.

Strictly decreasing prime quartets are A054804.

Strictly increasing prime quartets are A054819.

Equal prime quartets are A090832.

Weakly increasing prime quartets are A333383.

Weakly decreasing prime quartets are A333488.

Unequal prime quartets are A333490 (this sequence).

Partially unequal prime quartets are A333491.

Positions of adjacent equal prime gaps are A064113.

Positions of strict ascents in prime gaps are A258025.

Positions of strict descents in prime gaps are A258026.

Positions of adjacent unequal prime gaps are A333214.

Positions of weak ascents in prime gaps are A333230.

Positions of weak descents in prime gaps are A333231.

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

Programs

Mathematica

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