1,1

Primes prime(k) such that log(prime(k+1)-prime(k)) < log(log(prime(k)))-gamma, where log is the natural logarithm and gamma is Euler's constant (A001620).

Except for terms less than 41, A001359 (Lesser of twin primes) is a subsequence. From 41, the first term not included is 1279.

It has been conjectured that primes are distributed around their average spacing in a Poisson distribution (cf. D. A. Goldston and A. H. Ledoan). This is the basis of the conjecture that, for k tending to infinity, the asymptotic limit of the average of log(prime(k+1)-prime(k)) is log(log(prime(k))) - gamma (where gamma is Euler's constant). Also, the geometric mean of the gap between consecutive primes [p(k+1)-p(k)] is equivalent to log(prime(k)) / e^gamma.

D. A. Goldston and A. H. Ledoan, On the differences between consecutive prime numbers, I, arXiv:1111.3380 [math.NT], 2011-2012.

Carlos Rivera, Conjecture 82. Average of log Dn / log(logPn) equal R = 0,877 08..., The Prime Puzzles & Problems Connection.

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

29 is not a term because log(31-29) > log(log(29))-0.5772156649, i.e.: 0.693147 > 0.636894.

41 is a term because log(43-41) < log(log(41))-0.5772156649, i.e.: 0.693147 < 0.734779.

Select[Prime[Range[237]], Log[NextPrime[#]-#]<Log[Log[#]]-EulerGamma&] (* James C. McMahon, May 02 2025 *)

(PARI) forprime(P=3, 1500, my(Q=nextprime(P+1), LNDP=log(Q-P)); if(LNDP<log(log(P))-Euler, print1(P, ", ")));

Cf. A001359, A046869, A001223, A001620.

Sequence in context: A262722 A172406 A161613 * A345042 A210338 A139890

Adjacent sequences: A381041 A381042 A381043 * A381045 A381046 A381047

Alain Rocchelli, Apr 14 2025

approved