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A384618 Primes preceded and followed by gaps whose quotient (value greater or equal to 1) is greater than 2.

%I A384618 #12 Jun 11 2025 03:05:14
%S A384618 29,31,59,61,73,113,127,137,139,149,151,179,181,191,199,223,239,241,
%T A384618 269,271,283,307,317,331,347,419,421,431,433,467,521,523,541,569,571,
%U A384618 599,601,619,641,659,661,673,773,809,811,821,829,853,863,877,887,907,953,967
%N A384618 Primes preceded and followed by gaps whose quotient (value greater or equal to 1) is greater than 2.
%C A384618 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 A384618 Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 2/3.
%e A384618 19 is not a term because Max(19-17,23-19)/Min(19-17,23-19) = 4/2 = 2.
%e A384618 29 is a term because Max(29-23,31-29)/Min(29-23,31-29) = 6/2 = 3.
%e A384618 31 is a term because Max(31-29,37-31)/Min(31-29,37-31) = 6/2 = 3.
%e A384618 37 is not a term because Max(37-31,41-37)/Min(37-31,41-37) = 6/4 = 1.5.
%o A384618 (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 A384618 (Python)
%o A384618 from itertools import islice
%o A384618 from sympy import nextprime
%o A384618 def A384618_gen(): # generator of terms
%o A384618     p,q,r = 2,3,5
%o A384618     while True:
%o A384618         s, t = q-p, r-q
%o A384618         if s>(t<<1) or t>(s<<1): yield q
%o A384618         p, q, r = q, r, nextprime(r)
%o A384618 A384618_list = list(islice(A384618_gen(),54)) # _Chai Wah Wu_, Jun 10 2025
%Y A384618 Cf. A384603.
%K A384618 nonn
%O A384618 1,1
%A A384618 _Alain Rocchelli_, Jun 04 2025