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

1,1

Comments

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

Links

Table of n, a(n) for n=1..63.

Formula

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

Example

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.

Mathematica

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

Prog

(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<AR0, print1(P, ", ")));

Crossrefs

A288907 and A381850 are subsequences.

Cf. A083550.

Sequence in context: A289492 A262286 A108024 * A095081 A243437 A144709

Adjacent sequences: A383649 A383650 A383651 * A383653 A383654 A383655

Keyword

nonn

Author

Alain Rocchelli, May 04 2025

Status

approved