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A083371 Primes p such that q-p >= 8, where q is the next prime after p.

%I A083371 #18 Jan 26 2022 11:10:19
%S A083371 89,113,139,181,199,211,241,283,293,317,337,359,389,401,409,421,449,
%T A083371 467,479,491,509,523,547,577,619,631,661,683,691,701,709,719,743,761,
%U A083371 773,787,797,811,829,839,863,887,911,919,929,953,983,997,1021,1039,1051,1069
%N A083371 Primes p such that q-p >= 8, where q is the next prime after p.
%C A083371 The original definition by Cloitre was: [Start from any initial value F(1) >= 2 and define F(n) as the largest prime factor of F(1)+F(2)+F(3)+...+F(n-1). The sequence contains the primes satisfying F(2*p)=p supposed F(1)=7. Conjecture: F(n)= n/2+O(log n) and the sequence is infinite.] _Don Reble_ showed Jan 22 2022 that these are the same primes p followed by a prime gap of q-p >=8, where q is the next prime after p: [
%C A083371 Let X' be the first prime after X, 'X be the first prime before X.
%C A083371 The F sequence starting at "7" has 11 "7"s, then 6 "11"s, 6 "13"s, 6 "17"s, 6 "19"s, 10 "23"s, ...
%C A083371 One easily sees that the F sequence starting at prime S has S' instances of S; then for each prime P after S, it has (P'-'P) instances of P.  (A076973 is the F sequence starting at "2".)
%C A083371 The primes from S to P occupy the first [S' + (S''-S) + (S'''-S') + ... + (P' - 'P)] terms of F.
%C A083371 That sum telescopes to P'+P-S, and so
%C A083371     F(P'+P-S) = P;  F(P'+P-S+1) = P';
%C A083371     F(P+'P-S) = 'P; F(P+'P-S+1) = P.
%C A083371 If F(X) =P, then P+'P-S < X   <= P'+P-S.
%C A083371 If F(2P)=P, then P+'P-S < 2P  <= P'+P-S
%C A083371                      'P < P+S <= P'
%C A083371                             S <= P'-P
%C A083371 So this sequence has the primes P for which P'-P >= 7; and since P'-P is even (both primes are odd), P'-P >= 8. q.e.d.]
%H A083371 Nathaniel Johnston, <a href="/A083371/b083371.txt">Table of n, a(n) for n = 1..1500</a>
%H A083371 K. Soundararajan, <a href="https://doi.org/10.1090/S0273-0979-06-01142-6">Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim</a>, Bull. Amer. Math. Soc., 44 (2007), 1-18.
%H A083371 <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%F A083371 A000040 MINUS A124590. - _R. J. Mathar_, Jan 23 2022
%F A083371 A031926 UNION A031928 UNION A031930 UNION A031932 UNION ... - _R. J. Mathar_, Jan 23 2022
%p A083371 d:=8; M:=1000; t0:=[]; for n from 1 to M do p:=ithprime(n); if nextprime(p) - p >= d then t0:=[op(t0),p]; fi; od: t0; # _N. J. A. Sloane_, Dec 19 2006
%p A083371 f := proc(n) option remember: if(n=1)then return 7: fi: return max(op(numtheory[factorset](add(f(i),i=1..n-1)))): end: seq(`if`(f(2*ithprime(n))=ithprime(n),ithprime(n),NULL),n=1..200); # _Nathaniel Johnston_, Jun 25 2011, via Cloitre's F
%t A083371 Transpose[Select[Partition[Prime[Range[200]],2,1],Last[#]-First[#]>7&]][[1]] (* _Harvey P. Dale_, Jan 28 2013 *)
%Y A083371 Cf. A076973.
%K A083371 nonn
%O A083371 1,1
%A A083371 _Benoit Cloitre_, Jun 04 2003
%E A083371 Terms after a(20) from _Nathaniel Johnston_, Jun 26 2011
%E A083371 Merged with A124583 in response to Reble's seqfan post. - _R. J. Mathar_, Jan 24 2022