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.
FORMULA
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
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