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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
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.]
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
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