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A288907 Primes p whose distance from the next prime and from the previous prime is less than log(p).

%I A288907 #34 May 12 2025 17:21:32
%S A288907 71,101,103,107,109,193,197,227,229,281,311,313,349,433,439,443,461,
%T A288907 463,503,563,569,571,593,599,601,607,613,617,643,647,653,659,677,733,
%U A288907 739,757,823,827,857,859,881,883,941,947,971,977,1013,1019,1033,1063,1091,1093
%N A288907 Primes p whose distance from the next prime and from the previous prime is less than log(p).
%C A288907 Primes preceded and followed by less-than-average prime gaps (by the Prime Number Theorem, see link).
%C A288907 This sequence is a subsequence of A381850 and of A383652. - _Alain Rocchelli_, May 07 2025
%H A288907 Alois P. Heinz, <a href="/A288907/b288907.txt">Table of n, a(n) for n = 1..10000</a>
%H A288907 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeNumberTheorem.html">Prime Number Theorem</a>
%F A288907 A151800(a(n)) - log(a(n)) < a(n) < A151799(a(n)) + log(a(n)).
%F A288907 Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = (1-1/e)^2 (A068996). - _Alain Rocchelli_, May 07 2025
%e A288907 n = 23 is not a term because 23 - 19 > log(23) = 3.13...
%e A288907 n = 71 is a term because log(71) = 4.71.. and 73 - log(71) < 71 < 67 + log(71).
%p A288907 q:= p-> isprime(p) and is(max(nextprime(p)-p, p-prevprime(p))<log(p)):
%p A288907 select(q, [$3..1200])[];  # _Alois P. Heinz_, May 12 2025
%t A288907 Select[Range[2, 220] // Prime, Max[ Abs[# - NextPrime[#, {-1, 1}]]] < Log[#] &] (* _Giovanni Resta_, Jun 19 2017 *)
%o A288907 (Sage) [n for n in prime_range(3,1300) if next_prime(n)-n<log(n) and n-previous_prime(n)<log(n)] #
%o A288907 (PARI) is(n) = ispseudoprime(n) && n-precprime(n-1) < log(n) && nextprime(n+1)-n < log(n) \\ _Felix Fröhlich_, Jun 19 2017
%Y A288907 Cf. A151799, A151800, A288908, A068996, A381850, A383652.
%K A288907 nonn
%O A288907 1,1
%A A288907 _Giuseppe Coppoletta_, Jun 19 2017