cp's OEIS Frontend

A385093 Primes preceded and followed by gaps whose quotient (value greater or equal to 1) is greater than 2 and less than 3.

%I A385093 #13 Jun 29 2025 10:06:02
%S A385093 331,953,1381,1861,2161,2357,2371,2423,2879,3229,3271,3407,3491,3607,
%T A385093 3643,3691,3889,4057,4073,4139,4201,4507,4567,4751,4831,4903,4987,
%U A385093 5059,5153,5297,5309,5683,5897,6029,6053,6067,6173,6229,6529,6599,6653,6857,7079,7151,7159,7193,7283,7417,7717,7867,7963
%N A385093 Primes preceded and followed by gaps whose quotient (value greater or equal to 1) is greater than 2 and less than 3.
%C A385093 Primes prime(k) such that 2 < Max(prime(k)-prime(k-1), prime(k+1)-prime(k)) / Min(prime(k)-prime(k-1), prime(k+1)-prime(k)) < 3.
%F A385093 Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 1/6.
%e A385093 331 is a term because Max(331-317,337-331)/Min(331-317,337-331) = 14/6 = 2.3333.
%e A385093 953 is a term because Max(953-947,967-953)/Min(953-947,967-953) = 14/6 = 2.3333.
%e A385093 1381 is a term because Max(1381-1373,1399-1381)/Min(1381-1373,1399-1381) = 18/8 = 2.25.
%t A385093 Prime/@Select[Range[3,1000],2<Max[Prime[#]-Prime[#-1],Prime[#+1]-Prime[#]]/Min[Prime[#]-Prime[#-1],Prime[#+1]-Prime[#]]<3&] (* _James C. McMahon_, Jun 29 2025 *)
%o A385093 (PARI) forprime(P=3, 8000, my(M=P-precprime(P-1), Q=nextprime(P+1)-P, AR=max(M,Q)/min(M,Q)); if(AR>2 && AR<3, print1(P,", ")));
%Y A385093 Cf. A384603, A384618.
%K A385093 nonn
%O A385093 1,1
%A A385093 _Alain Rocchelli_, Jun 17 2025