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A270563 Integers k such that A086167(k) and A086168(k) are both prime.

%I A270563 #18 Jan 03 2020 10:31:50
%S A270563 1,15,45,105,135,231,807,1215,1329,1395,1593,1911,2301,2331,2493,3045,
%T A270563 3267,3417,3495,3897,4029,4059,4359,4377,4635,4665,4731,5265,6135,
%U A270563 6315,6429,6489,6795,6915,6999,7329,7515,7965,8469,8979,9183,9441,10755,11193,12039
%N A270563 Integers k such that A086167(k) and A086168(k) are both prime.
%C A270563 A013916 lists numbers n such that the sum of the first n primes is prime. With similar motivation, twin prime pairs generate prime pairs in this sequence. Note that 2*n also gives the difference between members of prime pair that is generated by sum of first n twin prime pairs.
%C A270563 First differences of this sequence are 14, 30, 60, 30, 96, 576, ...
%H A270563 Amiram Eldar, <a href="/A270563/b270563.txt">Table of n, a(n) for n = 1..10000</a>
%e A270563 15 is a term since A086167(15) = 1297 and A086168(15) = 1297 + 15*2 = 1327. 1297 and 1327 are both prime.
%t A270563 seq = {}; s1 = s2 = 0; c = n = 0; p = prv = 2; While[c < 45, p = NextPrime[p]; If[p == prv + 2, n++; s1 += prv; s2 += p; If[PrimeQ[s1] && PrimeQ[s2], c++; AppendTo[seq, n]]]; prv = p]; seq (* _Amiram Eldar_, Jan 03 2020 *)
%o A270563 (PARI) t(n, p=3) = {while( p+2 < (p=nextprime( p+1 )) || n-->0, ); p-2}
%o A270563 s1(n) = sum(k=1, n, t(k));
%o A270563 s2(n) = sum(k=1, n, t(k)+2);
%o A270563 for(n=1, 1e3, if(ispseudoprime(s1(n)) && ispseudoprime(s2(n)), print1(n, ", ")));
%Y A270563 Cf. A086167, A086168.
%K A270563 nonn
%O A270563 1,2
%A A270563 _Altug Alkan_, Mar 19 2016
%E A270563 More terms from _Amiram Eldar_, Jan 03 2020