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URL: https://oeis.org/A389650

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A389650
Minimal integer s such that the ceiling of the Golomb-Keller limit gives prime(n+1) (see comment).
1
2, 3, 4, 6, 6, 8, 7, 8, 13, 10, 14, 18, 14, 13, 16, 25, 18, 24, 29, 20, 25, 22, 23, 35, 44, 36, 45, 32, 21, 40, 37, 54, 31, 61, 40, 42, 51, 44, 49, 70, 41, 83, 66, 76, 34, 42, 78, 96, 72, 65, 93, 48, 65, 68, 75, 109, 74, 94, 109, 52, 52, 107, 130, 90, 52, 81, 74, 146, 106, 87, 79, 94, 99, 116, 95, 88, 117, 85, 86, 163, 90
OFFSET
1,1
COMMENTS
The Golomb-Keller formula expresses prime(n+1) as a limit: prime(n+1) = lim_{s->oo} (zeta(s)*Product_{j=1..n}(1-1/prime(j)^s) - 1)^(-1/s). Here a(n) is the smallest positive integer s such that ceiling(h(s)) = prime(n+1) where h(s) = (zeta(s)*Product_{j=1..n}(1-1/prime(j)^s) - 1)^(-1/s).
Peaks in a(n)/prime(n) correlate strongly with twin prime configurations.
LINKS
Benoit Cloitre, An effective analytic recurrence for prime numbers, arXiv:2508.02690v2 [math.NT], Oct. 2025.
S. W. Golomb, Formulas for the next prime, Pacific J. Math. 63, (1976).
J. B. Keller, A recursion equation for prime numbers, arXiv:0711.3940 [math.NT], 2007-2008.
FORMULA
a(n) >= 2 for all n >= 1.
a(n) <= prime(n) for all n >= 1 (using Nagura's theorem).
liminf_{n->oo} a(n)/prime(n) = 0 (by Westzynthius's theorem).
limsup_{n->oo} a(n)/prime(n) = C <= 0.4332 (using Dusart's bounds).
Under the twin prime conjecture, C > log(psi) = 0.3823, where psi = 1.46557 is the supergolden ratio (A092526).
For any c > 0.596, a(n) <= c*prime(n) for all sufficiently large n (proven). Conjectured: this holds for any c > C.
EXAMPLE
For n=6: The first 6 primes are 2, 3, 5, 7, 11, 13. We seek prime(7) = 17.
With s=7: h(7) = 15.856..., so ceiling(15.856) = 16 != 17.
With s=8: h(8) = 16.125..., so ceiling(16.125) = 17. Thus a(6) = 8.
PROG
(PARI) \\ Use: default(realprecision, 2000);
a(n) = {
my(pnext = prime(n+1), s = 2, P, Dn, h);
while(1,
P = 1.0;
for(j = 1, n, P *= (1.0 - 1.0/prime(j)^s));
Dn = zeta(s) * P;
h = (Dn - 1.0)^(-1.0/s);
if(ceil(h) == pnext, return(s));
s++;
if(s > prime(n), error("exceeded prime(n)"));
);
}
CROSSREFS
Cf. A000040 (primes), A001223 (prime gaps), A002110 (primorials), A092526 (supergolden ratio), A002386 (record prime gaps).
Sequence in context: A362813 A367964 A214965 * A350786 A385138 A134361
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Oct 09 2025
STATUS
approved