This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A382052 #27 Apr 16 2025 09:02:51
%S A382052 3,5,7,13,19,31,41,43,47,61,71,73,83,101,103,107,109,113,131,139,151,
%T A382052 167,181,193,197,199,227,229,233,241,271,281,283,311,313,317,337,349,
%U A382052 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
%N A382052 Primes prime(k) such that k*log(k)/prime(k) > (k-1)*log(k-1)/prime(k-1).
%C A382052 All terms of this sequence are contained in A060770.
%C A382052 a(n) ~ prime(round(n*e/(e-1))) as n tends to infinity, cf. A185393.
%F A382052 Limit_{n->oo} n / PrimePi(a(n)) = 1-1/e (A068996).
%e A382052 3 is a term because 2*log(2)/3 > 1*log(1)/2 and 3 is the 2nd prime following 2.
%e A382052 5 is a term because 3*log(3)/5 > 2*log(2)/3 and 5 is the 3rd prime following 3.
%t A382052 Select[Prime[Range[2, 109]],PrimePi[#]*Log[PrimePi[#]]/#>(PrimePi[#]-1)*Log[PrimePi[#]-1]/NextPrime[#,-1]&] (* _James C. McMahon_, Apr 14 2025 *)
%o A382052 (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,", ")));
%Y A382052 Cf. A382051, A060770, A068996, A185393.
%K A382052 nonn
%O A382052 1,1
%A A382052 _Alain Rocchelli_, Mar 13 2025