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A063713 Numbers n such that there exist primes p, q, r with n*2 = p - r = r + q (values of r are given in A063714).

%I A063713 #14 Jan 01 2024 14:07:37
%S A063713 4,5,6,7,8,9,10,11,12,13,15,16,17,18,20,21,22,23,24,25,27,28,30,32,33,
%T A063713 35,36,38,39,42,43,45,46,48,50,51,52,53,54,55,57,58,60,63,65,66,67,69,
%U A063713 70,71,72,75,77,78,80,81,84,85,87,88,90,93,96,97,98,99,100,101,102,105
%N A063713 Numbers n such that there exist primes p, q, r with n*2 = p - r = r + q (values of r are given in A063714).
%H A063713 Robert Israel, <a href="/A063713/b063713.txt">Table of n, a(n) for n = 1..10000</a>
%e A063713 10*2 = 20 = 23 - 3 = 3 + 17, A063714(7) = 3; 11*2 = 22 = 41 - 19 = 19 + 3, A063714(8) = 19 28 is missing because we have the prime sums (Goldbach): 5 + 23 = 11 + 17 and differences with primes less 28: 31 - 3 = 41 - 13 = 47 - 19; none of these have a prime in common.
%p A063713 filter:= proc(n) local k;
%p A063713   k:= 1;
%p A063713   while k < 2*n do
%p A063713     k:= nextprime(k);
%p A063713     if isprime(2*n+k) and isprime(2*n-k) then return true fi
%p A063713   od;
%p A063713   false
%p A063713 end proc:
%p A063713 select(filter, [$1..1000]); # _Robert Israel_, Oct 09 2017
%t A063713 okQ[n_] := AnyTrue[Prime[Range[PrimePi[2 n - 2]]], PrimeQ[2 n + #] && PrimeQ[2 n - #]&]; Select[Range[105], okQ] (* _Jean-François Alcover_, Feb 12 2018 *)
%Y A063713 Cf. A063714, A002373, A020483.
%K A063713 nonn
%O A063713 1,1
%A A063713 _Reinhard Zumkeller_, Aug 10 2001