A383652 Primes p preceded and followed by gaps whose product is less than (log(p))^2.

17, 19, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 139, 149, 151, 163, 167, 179, 181, 191, 193, 197, 199, 227, 229, 233, 239, 241, 269, 271, 277, 281, 283, 311, 313, 347, 349, 353, 379, 383, 397, 401, 419, 421, 431, 433, 439, 443, 457, 461, 463, 487, 491, 499, 503, 521, 523, 563, 569, 571, 593, 599

Offset: 1

Author

Alain Rocchelli, May 04 2025

Keywords

• nonn

Comments

Since the geometric mean is never greater than the arithmetic mean: A381850 is a subsequence.

Examples

			17 is a term because (17-13)*(19-17)=8 is less than (log(17))^2=8.0271.
19 is a term because (19-17)*(23-19)=8 is less than (log(19))^2=8.6697.
29 is not a term because(29-23)*(31-29)=12 is greater than (log(29))^2=11.3387.
		

Crossrefs

A288907 and A381850 are subsequences.

Cf. A083550.

Programs

Mathematica

Select[Range[2, 110] // Prime, (# - NextPrime[#, -1])(NextPrime[#] - #) < Log[#]^2 &] (* Stefano Spezia, May 04 2025 *)

PARI

forprime(P=3, 600, my(M=P-precprime(P-1), Q=nextprime(P+1)-P, AR=M*Q, AR0=(log(P))^2); if(AR
				

Formula

Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 0.720268...