A383216 Primes p which are preceded and followed by gaps whose difference is greater than 2*log(p).

113, 127, 523, 887, 907, 1087, 1129, 1151, 1277, 1327, 1361, 1669, 1693, 1931, 1951, 1973, 2203, 2311, 2333, 2477, 2557, 2971, 2999, 3163, 3251, 3299, 3469, 4049, 4297, 4327, 4523, 4547, 4783, 4861, 5119, 5147, 5237, 5351, 5381, 5531, 5557, 5591, 5749, 5779, 5981

Offset: 1

Author

Alain Rocchelli, Apr 19 2025

Keywords

• nonn

Comments

Primes prime(k) such that abs(prime(k-1)-2*prime(k)+prime(k+1)) > 2*log(prime(k)), where log is the natural logarithm.

Examples

			113 is a term because abs(109-2*113+127)=12 and 2*log(113)=9.4548.
127 is a term because abs(113-2*127+131)=10 and 2*log(127)=9.6884.
		

Crossrefs

Cf. A036263, A383215, A092553.

Programs

Mathematica

Select[Prime[Range[2,782]],Abs[NextPrime[#,-1]-2#+NextPrime[#]]>2Log[#]&] (* James C. McMahon, Apr 27 2025 *)

PARI

forprime(P=3, 6000, my(M=P-precprime(P-1), Q=nextprime(P+1)-P, AR1=min(M,Q), AR2=max(M,Q), AR0=2*log(P)); if(AR2-AR1>AR0, print1(P,", ")));

Formula

Limit_{n->oo} n / PrimePi(a(n)) = 1/e^2 (A092553).