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

%I A054804 #22 Jan 12 2023 13:03:15
%S A054804 31,61,89,211,271,293,449,467,607,619,709,743,839,863,919,1069,1291,
%T A054804 1409,1439,1459,1531,1637,1657,1669,1723,1759,1777,1831,1847,1861,
%U A054804 1979,1987,2039,2131,2311,2357,2371,2447,2459,2477,2503,2521,2557,2593,2633
%N A054804 First term of strong prime quartets: prime(m+1)-prime(m) > prime(m+2)-prime(m+1) > prime(m+3)-prime(m+2).
%C A054804 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 A054804 Robert Israel, <a href="/A054804/b054804.txt">Table of n, a(n) for n = 1..10000</a>
%F A054804 a(n) = prime(A335278(n)). - _Gus Wiseman_, May 31 2020
%e A054804 The first 10 strictly decreasing prime gap quartets:
%e A054804    31  37  41  43
%e A054804    61  67  71  73
%e A054804    89  97 101 103
%e A054804   211 223 227 229
%e A054804   271 277 281 283
%e A054804   293 307 311 313
%e A054804   449 457 461 463
%e A054804   467 479 487 491
%e A054804   607 613 617 619
%e A054804   619 631 641 643
%e A054804 For example, the primes (211,223,227,229) have differences (12,4,2), which are strictly decreasing, so 211 is in the sequence.
%e A054804 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 A054804 primes:= select(isprime,[seq(i,i=3..10000,2)]):
%p A054804 L:=  primes[2..-1]-primes[1..-2]:
%p A054804 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 A054804 ReplaceList[Array[Prime,100],{___,x_,y_,z_,t_,___}/;y-x>z-y>t-z:>x] (* _Gus Wiseman_, May 31 2020 *)
%t A054804 Select[Partition[Prime[Range[400]],4,1],Max[Differences[#,2]]<0&][[All,1]] (* _Harvey P. Dale_, Jan 12 2023 *)
%Y A054804 Cf. A051634, A054800-A054840.
%Y A054804 Prime gaps are A001223.
%Y A054804 Second prime gaps are A036263.
%Y A054804 All of the following use prime indices rather than the primes themselves:
%Y A054804 - Strictly decreasing prime gap quartets are A335278.
%Y A054804 - Strictly increasing prime gap quartets are A335277.
%Y A054804 - Equal prime gap quartets are A090832.
%Y A054804 - Weakly increasing prime gap quartets are A333383.
%Y A054804 - Weakly decreasing prime gap quartets are A333488.
%Y A054804 - Unequal prime gap quartets are A333490.
%Y A054804 - Partially unequal prime gap quartets are A333491.
%Y A054804 - Adjacent equal prime gaps are A064113.
%Y A054804 - Strict ascents in prime gaps are A258025.
%Y A054804 - Strict descents in prime gaps are A258026.
%Y A054804 - Adjacent unequal prime gaps are A333214.
%Y A054804 - Weak ascents in prime gaps are A333230.
%Y A054804 - Weak descents in prime gaps are A333231.
%Y A054804 Maximal weakly increasing intervals of prime gaps are A333215.
%Y A054804 Maximal strictly decreasing intervals of prime gaps are A333252.
%Y A054804 Cf. also A000040, A006560, A031217, A054800, A054805, A054806, A054807, A059044, A084758, A089180, A333253, A335278.
%K A054804 nonn
%O A054804 1,1
%A A054804 _Henry Bottomley_, Apr 10 2000