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A293621
Numbers k such that (2*k)^2 + 1 and (2*k+2)^2 + 1 are both primes.
1
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
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.
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.
Sequence in context: A350093 A177747 A288888 * A175879 A102371 A007230
KEYWORD
nonn
AUTHOR
Amiram Eldar, Oct 13 2017
STATUS
approved