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A263674 Double interprimes: a(n) = (q+r)/2 = (p+s)/2 with p consecutive primes.

%I A263674 #34 Jun 15 2025 18:53:16
%S A263674 9,12,15,18,30,42,60,81,102,105,108,120,144,165,186,195,228,260,270,
%T A263674 312,363,381,399,420,426,441,462,489,495,552,570,582,600,696,705,714,
%U A263674 765,816,825,858,870,882,897,924,987,1026,1050,1056,1092,1113,1167,1230
%N A263674 Double interprimes: a(n) = (q+r)/2 = (p+s)/2 with p<q<r<s consecutive primes.
%C A263674 Values of p (lesser of consecutive primes) are in the sequence A022885.
%H A263674 Amiram Eldar, <a href="/A263674/b263674.txt">Table of n, a(n) for n = 1..10000</a>
%H A263674 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Interprime.html">Interprime</a>
%e A263674 600 is in this sequence because 593, 599, 601 and 607 are consecutive primes, and 600 = (599+601)/2 = (593+607)/2.
%t A263674 (Prime@ # + Prime[# + 3])/2 & /@ Select[Range@ 240, (First@ # + Last@ #)/2 == (#[[2]] + #[[3]])/2 &@ Prime@ Range[#, # + 3] &] (* _Michael De Vlieger_, Nov 18 2015 *)
%t A263674 Mean/@Select[Partition[Prime[Range[300]],4,1],(#[[2]]+#[[3]])/2==(#[[1]]+#[[4]])/2&] (* _Harvey P. Dale_, Aug 18 2024 *)
%o A263674 (PARI) {forprime(q=3,2000,p=precprime(q-1); r=nextprime(q+1); s=nextprime(r+1);m=(q+r)/2;if(m==(p+s)/2,print1(m,", ")))}
%Y A263674 Cf. A022885, A075190, A075277, A075296, A078443, A130178, A263675, A263676.
%K A263674 nonn
%O A263674 1,1
%A A263674 _Antonio Roldán_, Oct 23 2015