A395565 - OEIS

%I #19 May 06 2026 08:50:29

%S 3,5,7,11,13,17,19,23,29,31,37,47,53,67,79,83,89,97,113,127,131,139,

%T 149,157,173,181,191,199,211,223,241,251,257,263,283,293,307,317,331,

%U 337,347,359,367,373,389,397,401,409,449,467,479,509,521,523,541,547

%N Primes p preceded and followed by primes whose difference is greater than 2*log(p).

%C Primes prime(k) such that prime(k+1) - prime(k-1) > 2*log(prime(k)).

%C Complement of A381850 within primes.

%F Conjecture: lim_{n->oo} n / pi(a(n)) = 3/e^2.

%e 37 is a term because 41-31=10 and 2*log(37)=7.2218.

%e 41 is not a term because 43-37=6 and 2*log(41)=7.4271.

%t q[k_]:=NextPrime[k]-NextPrime[k,-1]>2Log[k];Select[Prime[Range[2,101]],q] (* _James C. McMahon_, May 04 2026 *)

%o (PARI) forprime(P=3, 500, my(M=precprime(P-1), Q=nextprime(P+1)); if(Q-M>2*log(P), print1(P,", ")));

%Y Cf. A000720, A031131, A381850, A383216.

%K nonn

%O 1,1

%A _Alain Rocchelli_, Apr 28 2026