A054804 First term of strong prime quartets: prime(m+1)-prime(m) > prime(m+2)-prime(m+1) > prime(m+3)-prime(m+2).

31, 61, 89, 211, 271, 293, 449, 467, 607, 619, 709, 743, 839, 863, 919, 1069, 1291, 1409, 1439, 1459, 1531, 1637, 1657, 1669, 1723, 1759, 1777, 1831, 1847, 1861, 1979, 1987, 2039, 2131, 2311, 2357, 2371, 2447, 2459, 2477, 2503, 2521, 2557, 2593, 2633

Offset: 1

Author

Henry Bottomley, Apr 10 2000

Keywords

• nonn

Comments

Primes preceding the first member of pairs of consecutive primes in A051634 ("strong primes"), see example. (A051634 lists the middle member of the triplets, here we list the first member of the quadruplets.) - M. F. Hasler, Oct 27 2018, corrected thanks to Gus Wiseman, Jun 01 2020.

Examples

			The first 10 strictly decreasing prime gap 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, the primes (211,223,227,229) have differences (12,4,2), which are strictly decreasing, so 211 is in the sequence.
The second and third term of each quadruplet are consecutive terms in A051634: this is a characteristic property of this sequence. - _M. F. Hasler_, Jun 01 2020
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A051634, A054800-A054840.

Prime gaps are A001223.

Second prime gaps are A036263.

All of the following use prime indices rather than the primes themselves:

- Strictly decreasing prime gap quartets are A335278.

- Strictly increasing prime gap quartets are A335277.

- Equal prime gap quartets are A090832.

- Weakly increasing prime gap quartets are A333383.

- Weakly decreasing prime gap quartets are A333488.

- Unequal prime gap quartets are A333490.

- Partially unequal prime gap quartets are A333491.

- Adjacent equal prime gaps are A064113.

- Strict ascents in prime gaps are A258025.

- Strict descents in prime gaps are A258026.

- Adjacent unequal prime gaps are A333214.

- Weak ascents in prime gaps are A333230.

- Weak descents in prime gaps are A333231.

Maximal weakly increasing intervals of prime gaps are A333215.

Maximal strictly decreasing intervals of prime gaps are A333252.

Cf. also A000040, A006560, A031217, A054800, A054805, A054806, A054807, A059044, A084758, A089180, A333253, A335278.

Programs

Maple

primes:= select(isprime,[seq(i,i=3..10000,2)]):
L:=  primes[2..-1]-primes[1..-2]:
primes[select(t -> L[t+2] < L[t+1] and L[t+1] < L[t], [$1..nops(L)-2])]; # Robert Israel, Jun 28 2018

Mathematica

ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-x>z-y>t-z:>x] (* Gus Wiseman, May 31 2020 *)
Select[Partition[Prime[Range[400]],4,1],Max[Differences[#,2]]<0&][[All,1]] (* Harvey P. Dale, Jan 12 2023 *)

Formula

a(n) = prime(A335278(n)). - Gus Wiseman, May 31 2020