A382051 Primes prime(k) such that k*log(k)/prime(k) < (k-1)*log(k-1)/prime(k-1).

11, 17, 23, 29, 37, 53, 59, 67, 79, 89, 97, 127, 137, 149, 157, 163, 173, 179, 191, 211, 223, 239, 251, 257, 263, 269, 277, 293, 307, 331, 347, 367, 397, 409, 419, 431, 457, 479, 487, 499, 521, 541, 557, 587, 631, 641, 673, 691, 701, 709, 719, 727, 751, 769, 787, 797

Offset: 1

Author

Alain Rocchelli, Mar 13 2025

Keywords

• nonn

Comments

a(n) ~ prime(round(n*e)) as n tends to infinity, where e is Euler's number.

Examples

			11 is a term because 5*log(5)/11 < 4*log(4)/7 and 11 is the 5th prime following 7.
17 is a term because 7*log(7)/17 < 6*log(6)/13 and 17 is the 7th prime following 13.
		

Crossrefs

A subsequence is A060769.

Cf. A382052, A060769, A001113, A068985.

Programs

Mathematica

Select[Prime[Range[2,139]],PrimePi[#]*Log[PrimePi[#]]/#<(PrimePi[#]-1)*Log[PrimePi[#]-1]/NextPrime[#,-1]&] (* James C. McMahon, Apr 08 2025 *)

PARI

my(N=1); forprime(P=3, 800, my(Q=precprime(P-1), AR0=N*log(N)/Q, AR=(N+1)*log(N+1)/P); N++; if(AR
				

Formula

Limit_{n->oo} n / PrimePi(a(n)) = 1/e.