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

89, 113, 139, 181, 199, 211, 241, 283, 293, 317, 337, 359, 389, 401, 409, 421, 449, 467, 479, 491, 509, 523, 547, 577, 619, 631, 661, 683, 691, 701, 709, 719, 743, 761, 773, 787, 797, 811, 829, 839, 863, 887, 911, 919, 929, 953, 983, 997, 1021, 1039, 1051, 1069

Offset: 1

Author

Benoit Cloitre, Jun 04 2003

Keywords

• nonn

Comments

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: [

Let X' be the first prime after X, 'X be the first prime before X.

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, ...

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".)

The primes from S to P occupy the first [S' + (S''-S) + (S'''-S') + ... + (P' - 'P)] terms of F.

That sum telescopes to P'+P-S, and so

F(P'+P-S) = P; F(P'+P-S+1) = P';

F(P+'P-S) = 'P; F(P+'P-S+1) = P.

If F(X) =P, then P+'P-S < X <= P'+P-S.

If F(2P)=P, then P+'P-S < 2P <= P'+P-S

'P < P+S <= P'

S <= P'-P

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.]

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..1500
• K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18.
• Index entries for primes, gaps between

Crossrefs

Cf. A076973.

Programs

Maple

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
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

Mathematica

Transpose[Select[Partition[Prime[Range[200]],2,1],Last[#]-First[#]>7&]][[1]] (* Harvey P. Dale, Jan 28 2013 *)

Formula

A000040 MINUS A124590. - R. J. Mathar, Jan 23 2022

A031926 UNION A031928 UNION A031930 UNION A031932 UNION ... - R. J. Mathar, Jan 23 2022

Extensions

Terms after a(20) from Nathaniel Johnston, Jun 26 2011

Merged with A124583 in response to Reble's seqfan post. - R. J. Mathar, Jan 24 2022