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

1, 2, 7, 12, 27, 62, 102, 192, 232, 317, 322, 357, 547, 572, 587, 622, 637, 657, 687, 782, 807, 837, 842, 982, 1027, 1042, 1047, 1202, 1227, 1267, 1332, 1417, 1462, 1567, 1652, 1767, 1877, 1887, 2012, 2077, 2087, 2182, 2302, 2307, 2367, 2392, 2397, 2477, 2507

Offset

1,2

Comments

Sierpiński proved that under Schinzel's hypothesis H this sequence is infinite. He gives the 24 terms below 10^3.

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.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Wacław Sierpiński, Remarque sur la distribution de nombres premiers, Matematički Vesnik, Vol. 2(17), Issue 31 (1965), pp. 77-78.

Eric Weisstein's World of Mathematics, Near-Square Prime.

Wikipedia, Schinzel's hypothesis H.

Formula

a(n) = A096012(n)/2. - Amiram Eldar, Feb 24 2020

Example

1 is in the sequence since (2*1)^2 + 1 = 5 and (2*1+2)^2 + 1 = 17 are both primes.

Mathematica

Select[Range[10^4], AllTrue[{(2#)^2+1, (2#+2)^2+1}, PrimeQ] &]

Prog

(PARI) isok(n) = isprime((2*n)^2 + 1) && isprime((2*n+2)^2 + 1); \\ Michel Marcus, Oct 13 2017

Crossrefs

Subsequence of A001912.

Cf. A002496, A002522, A005574, A053755, A096012.

Sequence in context: A350093 A177747 A288888 * A175879 A102371 A007230

Adjacent sequences: A293618 A293619 A293620 * A293622 A293623 A293624

Keyword

nonn

Author

Amiram Eldar, Oct 13 2017

Status

approved