A335277 First index of strictly increasing prime quartets.

7, 13, 22, 28, 49, 60, 64, 69, 70, 75, 78, 85, 89, 95, 104, 116, 122, 123, 144, 148, 152, 155, 173, 178, 182, 195, 201, 206, 212, 215, 219, 225, 226, 230, 236, 237, 244, 253, 256, 257, 265, 288, 302, 307, 315, 325, 328, 329, 332, 333, 336, 348, 355, 361, 373

Offset: 1

Author

Gus Wiseman, May 30 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 strictly increasing prime quartets:
   17  19  23  29
   41  43  47  53
   79  83  89  97
  107 109 113 127
  227 229 233 239
  281 283 293 307
  311 313 317 331
  347 349 353 359
  349 353 359 367
  379 383 389 397
For example, 107 is the 28th prime, and the primes (107,109,113,127) have differences (2,4,14), which are strictly increasing, so 28 is in the sequence.
		

Crossrefs

Prime gaps are A001223.

Second prime gaps are A036263.

Strictly decreasing prime quartets are A335278.

Equal prime quartets are A090832.

Weakly increasing prime quartets are A333383.

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.

Lengths of maximal weakly decreasing sequences of prime gaps are A333212.

Lengths of maximal strictly increasing sequences of prime gaps are A333253.

Cf. A000040, A006560, A031217, A054800, A054819, A059044, A084758, A089180.

Programs

Mathematica

ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-xPrimePi[x]]

Formula

prime(a(n)) = A054819(n).