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A293621 Numbers k such that (2*k)^2 + 1 and (2*k+2)^2 + 1 are both primes.

%I A293621 #22 Feb 16 2025 08:33:51
%S A293621 1,2,7,12,27,62,102,192,232,317,322,357,547,572,587,622,637,657,687,
%T A293621 782,807,837,842,982,1027,1042,1047,1202,1227,1267,1332,1417,1462,
%U A293621 1567,1652,1767,1877,1887,2012,2077,2087,2182,2302,2307,2367,2392,2397,2477,2507
%N A293621 Numbers k such that (2*k)^2 + 1 and (2*k+2)^2 + 1 are both primes.
%C A293621 Sierpiński proved that under Schinzel's hypothesis H this sequence is infinite. He gives the 24 terms below 10^3.
%C A293621 Sierpiński noted that the only triple of consecutive primes of the form (2n)^2 + 1 are for n = 1 (i.e., 1 and 2 are the only consecutive terms in this sequence), since every triple of consecutive terms contains at least one term which is divisible by 5.
%H A293621 Amiram Eldar, <a href="/A293621/b293621.txt">Table of n, a(n) for n = 1..10000</a>
%H A293621 Wacław Sierpiński, <a href="http://gdz.sub.uni-goettingen.de/en/dms/load/img/?PPN=PPN311571026_0017&amp;DMDID=dmdlog25">Remarque sur la distribution de nombres premiers</a>, Matematički Vesnik, Vol. 2(17), Issue 31 (1965), pp. 77-78.
%H A293621 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime</a>.
%H A293621 Wikipedia, <a href="https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H">Schinzel's hypothesis H</a>.
%F A293621 a(n) = A096012(n)/2. - _Amiram Eldar_, Feb 24 2020
%e A293621 1 is in the sequence since (2*1)^2 + 1 = 5 and (2*1+2)^2 + 1 = 17 are both primes.
%t A293621 Select[Range[10^4], AllTrue[{(2#)^2+1, (2#+2)^2+1}, PrimeQ] &]
%o A293621 (PARI) isok(n) = isprime((2*n)^2 + 1) && isprime((2*n+2)^2 + 1); \\ _Michel Marcus_, Oct 13 2017
%Y A293621 Subsequence of A001912.
%Y A293621 Cf. A002496, A002522, A005574, A053755, A096012.
%K A293621 nonn
%O A293621 1,2
%A A293621 _Amiram Eldar_, Oct 13 2017