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A384603 Primes preceded and followed by gaps whose quotient (value greater or equal to 1) is less than 2.

%I A384603 #10 Jun 11 2025 03:19:59
%S A384603 5,23,37,47,53,67,79,83,89,131,157,163,167,173,211,233,251,257,263,
%T A384603 277,293,337,353,359,367,373,379,383,389,409,439,443,449,479,503,547,
%U A384603 557,563,577,587,593,607,613,631,647,653,677,683,691,701,709,719,727,733,739,751,757,787,797
%N A384603 Primes preceded and followed by gaps whose quotient (value greater or equal to 1) is less than 2.
%C A384603 Primes prime(k) such that Max(prime(k)-prime(k-1),prime(k+1)-prime(k)) / Min(prime(k)-prime(k-1),prime(k+1)-prime(k)) < 2.
%F A384603 Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 1/3.
%e A384603 5 is a term because Max(5-3,7-5)/Min(5-3,7-5) = 2/2 = 1.
%e A384603 23 is a term because Max(23-19,29-23)/Min(23-19,29-23) = 6/4 = 1.5.
%e A384603 37 is a term because Max(37-31,41-37)/Min(37-31,41-37) = 6/4 = 1.5.
%o A384603 (PARI) forprime(P=3, 1000, my(M=P-precprime(P-1), Q=nextprime(P+1)-P, AR=max(M,Q)/min(M,Q), AR0=2); if(AR<AR0, print1(P,", ")));
%o A384603 (Python)
%o A384603 from itertools import islice
%o A384603 from sympy import nextprime
%o A384603 def A384603_gen(): # generator of terms
%o A384603     p,q,r = 2,3,5
%o A384603     while True:
%o A384603         s, t = q-p, r-q
%o A384603         if s<(t<<1) and t<(s<<1): yield q
%o A384603         p, q, r = q, r, nextprime(r)
%o A384603 A384603_list = list(islice(A384603_gen(),59)) # _Chai Wah Wu_, Jun 10 2025
%Y A384603 Cf. A383215.
%K A384603 nonn
%O A384603 1,1
%A A384603 _Alain Rocchelli_, Jun 04 2025