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A083371 Primes p such that q-p >= 8, where q is the next prime after p. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

FORMULA

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

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

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 *)

CROSSREFS

Cf. A076973.

Sequence in context: A350724 A121608 A135144 * A124583 A257843 A155106

Adjacent sequences:  A083368 A083369 A083370 * A083372 A083373 A083374

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Jun 04 2003

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

STATUS

approved

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Last modified May 20 13:02 EDT 2022. Contains 353873 sequences. (Running on oeis4.)