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%I #15 Nov 10 2022 07:43:21
%S 7,60,72,114,186,378,474,480,762,884,1266,1338,1374,2004,2742,3012,
%T 3234,3246,3276,3282,3618,3936,4230,4620,5154,5514,5544,5724,5886,
%U 6006,6054,6084,6234,6306,6414,6510,6522,6762,6774,6858,7410,7422,7764,8286,8394,8538,8688,8826,8892,8916,9186,9264
%N Numbers k such that there exists a pair of primes (p,q) with p+q = k such that p*q + k, p*q - k, p*q + A001414(k) and p*q - A001414(k) are all prime.
%C Most terms == 0 (mod 6), but there are terms such as 7 and 15585 where p = 2, and
%C 884 and 10010 where p = 3.
%C Conjecture: the sequence includes all sufficiently large k such that both k and A001414(k) are divisible by 6.
%H Robert Israel, <a href="/A358132/b358132.txt">Table of n, a(n) for n = 1..828</a>
%e a(5) = 186 is a term because p = 89 and q = 97 are primes with 89 + 97 = 186, 186 = 2*3*31 so A001414(186) = 2+3+31 = 36, and 89*97 + 186 = 8819, 89*97 - 186 = 8447, 89*97 + 36 = 8669 and 89*97 - 36 = 8597 are all primes.
%p spf:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:
%p filter:= proc(k) local p,q,pq,s;
%p s:= spf(k);
%p if k::odd then
%p q:= k-2;
%p return (s::odd and isprime(q) and isprime(2*q+s) and isprime(2*q-s) and isprime(2*q+k) and isprime(2*q-k))
%p fi;
%p if k mod 6 <> 0 then
%p q:= k-3;
%p return (isprime(q) and isprime(3*q+s) and isprime(3*q-s) and isprime(3*q+k) and isprime(3*q-k))
%p fi;
%p p:= 2;
%p do
%p p:= nextprime(p);
%p q:= k-p;
%p if p > q then return false fi;
%p pq:= p*q;
%p if isprime(q) and isprime(pq+k) and isprime(pq-k) and isprime(pq+s) and isprime(pq-s) then return true fi
%p od
%p end proc:
%p select(filter, [$1..10000]);
%Y Cf. A001414.
%K nonn
%O 1,1
%A _J. M. Bergot_ and _Robert Israel_, Oct 31 2022
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