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First term of strong prime quartets: prime(m+1)-prime(m) > prime(m+2)-prime(m+1) > prime(m+3)-prime(m+2).
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%I #22 Jan 12 2023 13:03:15

%S 31,61,89,211,271,293,449,467,607,619,709,743,839,863,919,1069,1291,

%T 1409,1439,1459,1531,1637,1657,1669,1723,1759,1777,1831,1847,1861,

%U 1979,1987,2039,2131,2311,2357,2371,2447,2459,2477,2503,2521,2557,2593,2633

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

%C 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.

%H Robert Israel, <a href="/A054804/b054804.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = prime(A335278(n)). - _Gus Wiseman_, May 31 2020

%e The first 10 strictly decreasing prime gap quartets:

%e 31 37 41 43

%e 61 67 71 73

%e 89 97 101 103

%e 211 223 227 229

%e 271 277 281 283

%e 293 307 311 313

%e 449 457 461 463

%e 467 479 487 491

%e 607 613 617 619

%e 619 631 641 643

%e For example, the primes (211,223,227,229) have differences (12,4,2), which are strictly decreasing, so 211 is in the sequence.

%e 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

%p primes:= select(isprime,[seq(i,i=3..10000,2)]):

%p L:= primes[2..-1]-primes[1..-2]:

%p primes[select(t -> L[t+2] < L[t+1] and L[t+1] < L[t], [$1..nops(L)-2])]; # _Robert Israel_, Jun 28 2018

%t ReplaceList[Array[Prime,100],{___,x_,y_,z_,t_,___}/;y-x>z-y>t-z:>x] (* _Gus Wiseman_, May 31 2020 *)

%t Select[Partition[Prime[Range[400]],4,1],Max[Differences[#,2]]<0&][[All,1]] (* _Harvey P. Dale_, Jan 12 2023 *)

%Y Cf. A051634, A054800-A054840.

%Y Prime gaps are A001223.

%Y Second prime gaps are A036263.

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

%Y - Strictly decreasing prime gap quartets are A335278.

%Y - Strictly increasing prime gap quartets are A335277.

%Y - Equal prime gap quartets are A090832.

%Y - Weakly increasing prime gap quartets are A333383.

%Y - Weakly decreasing prime gap quartets are A333488.

%Y - Unequal prime gap quartets are A333490.

%Y - Partially unequal prime gap quartets are A333491.

%Y - Adjacent equal prime gaps are A064113.

%Y - Strict ascents in prime gaps are A258025.

%Y - Strict descents in prime gaps are A258026.

%Y - Adjacent unequal prime gaps are A333214.

%Y - Weak ascents in prime gaps are A333230.

%Y - Weak descents in prime gaps are A333231.

%Y Maximal weakly increasing intervals of prime gaps are A333215.

%Y Maximal strictly decreasing intervals of prime gaps are A333252.

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

%K nonn

%O 1,1

%A _Henry Bottomley_, Apr 10 2000