A382052 Primes prime(k) such that k*log(k)/prime(k) > (k-1)*log(k-1)/prime(k-1).

3, 5, 7, 13, 19, 31, 41, 43, 47, 61, 71, 73, 83, 101, 103, 107, 109, 113, 131, 139, 151, 167, 181, 193, 197, 199, 227, 229, 233, 241, 271, 281, 283, 311, 313, 317, 337, 349, 353, 359, 373, 379, 383, 389, 401, 421, 433, 439, 443, 449, 461, 463, 467, 491, 503, 509, 523, 547, 563, 569, 571, 577, 593, 599

Offset: 1

Author

Alain Rocchelli, Mar 13 2025

Keywords

• nonn

Comments

All terms of this sequence are contained in A060770.

a(n) ~ prime(round(n*e/(e-1))) as n tends to infinity, cf. A185393.

Examples

			3 is a term because 2*log(2)/3 > 1*log(1)/2 and 3 is the 2nd prime following 2.
5 is a term because 3*log(3)/5 > 2*log(2)/3 and 5 is the 3rd prime following 3.
		

Crossrefs

Cf. A382051, A060770, A068996, A185393.

Programs

Mathematica

Select[Prime[Range[2, 109]],PrimePi[#]*Log[PrimePi[#]]/#>(PrimePi[#]-1)*Log[PrimePi[#]-1]/NextPrime[#,-1]&] (* James C. McMahon, Apr 14 2025 *)

PARI

my(N=1); forprime(P=3, 600, my(Q=precprime(P-1), AR0=N*log(N)/Q, AR=(N+1)*log(N+1)/P); N++; if(AR>AR0, print1(P,", ")));

Formula

Limit_{n->oo} n / PrimePi(a(n)) = 1-1/e (A068996).