A381044 Primes prime(k) followed by a gap, prime(k+1)-prime(k), smaller than the local geometric average gap between consecutive primes: log(prime(k))/e^(gamma).

41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1279, 1289, 1297, 1301, 1303, 1319, 1423, 1427, 1429, 1447, 1451, 1481, 1483, 1487, 1489

Offset: 1

Author

Alain Rocchelli, Apr 14 2025

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A001359, A046869, A001223, A001620.

Programs

Mathematica

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

PARI

forprime(P=3, 1500, my(Q=nextprime(P+1), LNDP=log(Q-P)); if(LNDP
				

Formula

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