A333383 First index of weakly increasing prime quartets.

1, 2, 7, 13, 14, 22, 28, 35, 38, 45, 49, 54, 60, 64, 69, 70, 75, 78, 85, 89, 95, 104, 109, 116, 117, 122, 123, 144, 148, 152, 155, 159, 160, 163, 164, 173, 178, 182, 183, 184, 187, 194, 195, 198, 201, 206, 212, 215, 218, 219, 225, 226, 230, 236, 237, 238, 244

Offset: 1

Author

Gus Wiseman, May 14 2020

Keywords

• nonn

Comments

Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k) <= g(k + 1) <= g(k + 2).

Examples

			The first 10 weakly increasing prime quartets:
    2   3   5   7
    3   5   7  11
   17  19  23  29
   41  43  47  53
   43  47  53  59
   79  83  89  97
  107 109 113 127
  149 151 157 163
  163 167 173 179
  197 199 211 223
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.
		

Crossrefs

Prime gaps are A001223.

Second prime gaps are A036263.

Strictly decreasing prime quartets are A054804.

Strictly increasing prime quartets are A054819.

Equal prime quartets are A090832.

Weakly increasing prime quartets are A333383 (this sequence).

Weakly decreasing prime quartets are A333488.

Unequal prime quartets are A333490.

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.

Indices of weakly increasing rows of A066099 are A225620.

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

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

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

Programs

Mathematica

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