A333488 First index of weakly decreasing prime quartets.

11, 15, 18, 24, 36, 39, 46, 47, 53, 54, 55, 58, 62, 72, 73, 87, 91, 101, 102, 106, 107, 110, 111, 114, 118, 127, 128, 129, 132, 146, 150, 157, 180, 186, 193, 199, 210, 217, 223, 228, 232, 239, 242, 259, 260, 263, 269, 270, 271, 274, 275, 282, 283, 284, 290

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) >= g(k + 2).

Examples

			The first 10 weakly decreasing prime quartets:
   31  37  41  43
   47  53  59  61
   61  67  71  73
   89  97 101 103
  151 157 163 167
  167 173 179 181
  199 211 223 227
  211 223 227 229
  241 251 257 263
  251 257 263 269
For example, 241 is the 53rd prime, and the primes (241,251,257,263) have differences (10,6,6), which are weakly decreasing, so 53 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.

Weakly decreasing prime quartets are A333488 (this sequence).

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 decreasing rows of A066099 are A114994.

Lengths of maximal weakly decreasing subsequences of prime gaps: A333212.

Lengths of maximal strictly increasing subsequences of prime gaps: A333253.

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

Programs

Mathematica

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