A381323 - OEIS

%I #14 Mar 02 2025 23:55:22

%S 11,37,37,59,59,59,79,79,137,151,229,229,347,367,373,379,379,397,397,

%T 571,571,571,587,587,587,587,587,587,587,587,853,853,853,877,877,877,

%U 877,877,967,967,1009,1009,1019,1021,1021,1277,1297,1297,1361,1361,1361,1361,1361,1361,1361

%N 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).

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

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

%e 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.. .

%e 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.. .

%e 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.

%o (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())));

%Y Cf. A381293.

%K nonn

%O 1,1

%A _Alain Rocchelli_, Feb 21 2025