A089180 -id:A089180 - cp's OEIS frontend

A335278 First index of strictly decreasing prime quartets.

11, 18, 24, 47, 58, 62, 87, 91, 111, 114, 127, 132, 146, 150, 157, 180, 210, 223, 228, 232, 242, 259, 260, 263, 269, 274, 275, 282, 283, 284, 299, 300, 309, 321, 344, 350, 351, 363, 364, 367, 368, 369, 375, 378, 382, 388, 393, 399, 406, 409, 413, 431, 442, 446

Offset: 1

Gus Wiseman, May 30 2020

nonn

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

			The first 10 strictly decreasing prime quartets:
   31  37  41  43
   61  67  71  73
   89  97 101 103
  211 223 227 229
  271 277 281 283
  293 307 311 313
  449 457 461 463
  467 479 487 491
  607 613 617 619
  619 631 641 643
For example, 211 is the 47th prime, and the primes (211,223,227,229) have differences (12,4,2), which are strictly decreasing, so 47 is in the sequence.

Prime gaps are A001223.
Second prime gaps are A036263.
Strictly increasing prime quartets are A335277.
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.
Indices of strictly decreasing rows of A066099 are A333256.
Lengths of maximal weakly increasing sequences of prime gaps are A333215.
Lengths of maximal strictly decreasing sequences of prime gaps are A333252.
Cf. A000040, A006560, A031217, A054800, A054804, A059044, A084758, A089180, A333253.

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

prime(a(n)) = A054804(n).

A373817 Positions of terms > 1 in the run-lengths of the first differences of the odd primes.

Original entry on oeis.org

2, 14, 34, 36, 42, 49, 66, 94, 98, 100, 107, 117, 147, 150, 169, 171, 177, 181, 199, 219, 250, 268, 315, 333, 361, 392, 398, 435, 477, 488, 520, 565, 570, 585, 592, 595, 628, 642, 660, 666, 688, 715, 744, 765, 772, 778, 829, 842, 897, 906, 931, 932, 961, 1025

Offset: 1

Gus Wiseman, Jun 23 2024

nonn

Positions of terms > 1 in A333254. In other words, the a(n)-th run of differences of odd primes has length > 1.

			Primes 54 to 57 are {251, 257, 263, 269}, with differences (6,6,6). This is the 49th run, and the first of length > 2.

Positions of adjacent equal prime gaps are A064113.
Positions of adjacent unequal prime gaps are A333214.
Positions of terms > 1 in A333254, run-lengths A373821, firsts A335406.
A000040 lists the primes, differences A001223.
A027833 gives antirun lengths of odd primes, run-lengths A373820.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
Cf. A006560, A006562, A029707, A031217, A037201, A038664, A069010, A073051, A089180, A251092.

Join@@Position[Length /@ Split[Differences[Select[Range[1000],PrimeQ]]] // Most,x_Integer?(#>1&)]

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A335278 First index of strictly decreasing prime quartets.

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