A381293 a(n) is the least prime p such that the number of primes between p (not inclusive) and p+n*log(p) is n.

11, 37, 37, 59, 59, 59, 79, 79, 71, 67, 67, 179, 137, 131, 127, 227, 223, 191, 347, 349, 349, 337, 331, 331, 347, 347, 347, 307, 557, 557, 431, 557, 557, 547, 541, 547, 541, 431, 811, 797, 569, 821, 809, 811, 797, 797, 797, 797, 797, 797

Offset

1,1

Comments

a(n) ~ n^(3/2) as n -> oo.

Links

Table of n, a(n) for n=1..50.

Formula

Limit_{N->oo} (Sum_{n=2..N} log(a(n))) / (Sum_{n=2..N} log(n)) = 3/2.

Example

a(1) = 11 since 11 is the smallest prime for which we find the only prime 13 between 11 and 11+1*log(11)=13.39.. .
a(2) = 37 since 37 is the smallest prime for which we find the two primes 41 and 43 between 37 and 37+2*log(37)=44.22.. .

Prog

(PARI) for(K=1, 50, forprime(P=2, 50^2, AR=P+K*log(P); NPR=primepi(AR)-primepi(P); if(NPR==K, print1(P, ", "); break())));
(PARI) a(n) = my(p=2); while (primepi(p+n*log(p)) - primepi(p) != n, p=nextprime(p+1)); p; \\ Michel Marcus, Feb 21 2025

Crossrefs

Sequence in context: A306498 A195201 A233403 * A381323 A054293 A072859

Adjacent sequences: A381290 A381291 A381292 * A381294 A381295 A381296

Keyword

nonn,new

Author

Alain Rocchelli, Feb 19 2025

Status

approved