A381323 a(n) is the least nondecreasing prime p such that n is the number of primes between p (not included) and p+n*log(p).

11, 37, 37, 59, 59, 59, 79, 79, 137, 151, 229, 229, 347, 367, 373, 379, 379, 397, 397, 571, 571, 571, 587, 587, 587, 587, 587, 587, 587, 587, 853, 853, 853, 877, 877, 877, 877, 877, 967, 967, 1009, 1009, 1019, 1021, 1021, 1277, 1297, 1297, 1361, 1361, 1361, 1361, 1361, 1361, 1361

Offset

1,1

Links

Table of n, a(n) for n=1..55.

Formula

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

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 first 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 second prime for which we find the two primes 41 and 43 between 37 and 37+2*log(37)=44.22.. .
a(9) = 137 since 137 is the 9th prime (>= a(8)) for which we find 9 primes between 137 and 137+9*log(137)~181.29.

Prog

(PARI) Pdeb=2; for(K=1, 40, forprime(P=Pdeb, 40^2, AR=P+K*log(P); NPR=primepi(AR)-primepi(P); if(NPR==K, print1(P, ", "); Pdeb=P; break())));

Crossrefs

Cf. A381293.

Sequence in context: A195201 A233403 A381293 * A054293 A072859 A125744

Adjacent sequences: A381320 A381321 A381322 * A381326 A381327 A381328

Keyword

nonn,new

Author

Alain Rocchelli, Feb 21 2025

Status

approved