A217993
Smallest k such that k^(2^n) + 1 and (k+2)^(2^n) + 1 are both prime.
Original entry on oeis.org
2, 2, 2, 2, 74, 112, 2162, 63738, 13220, 54808, 3656570, 6992032, 125440, 103859114, 56414914, 87888966
Offset: 0
a(0) = 2 because 2^1+1 = 3 and 4^1+1 = 5 are prime;
a(1) = 2 because 2^2+1 = 5 and 4^2+1 = 17 are prime;
a(2) = 2 because 2^4+1 = 17 and 4^4+1 = 257 are prime;
a(3) = 2 because 2^8+1 = 257 and 4^8+1 = 65537 are prime.
Cf.
A006313,
A006314,
A006315,
A006316,
A056994,
A056995,
A057465,
A057002,
A088361,
A088362,
A118539.
-
for n from 0 to 5 do:ii:=0:for k from 2 by 2 to 10000 while(ii=0) do:if type(k^(2^n)+1,prime)=true and type((k+2)^(2^n)+1,prime)=true then ii:=1: printf ( "%d %d \n",n,k):else fi:od:od:
A272137
Primes of the form k^16 + 1.
Original entry on oeis.org
2, 65537, 197352587024076973231046657, 808551180810136214718004658177, 1238846438084943599707227160577, 37157429083410091685945089785857, 123025056645280288014028950372089857, 150838912030874130174020868290707457
Offset: 1
Cf. Sequences of numbers n such that n^(2^k)+1 is a prime p for k = 1-13:
A005574 (k=1),
A000068 (k=2),
A006314 (k=3),
A006313 (k=4),
A006315 (k=5),
A006316 (k=6),
A056994 (k=7),
A056995 (k=8),
A057465 (k=9),
A057002 (k=10),
A088361 (k=11),
A088362 (k=12),
A226528 (k=13).
-
[n^16 + 1: n in [1..700] | IsPrime(n^16 + 1)];
-
A272137:=n->`if`(isprime(n^16+1), n^16+1, NULL): seq(A272137(n), n=1..200); # Wesley Ivan Hurt, May 11 2016
A277967
Number of even numbers b with 0 < b < 2^n such that b^(2^n) + 1 is prime.
Original entry on oeis.org
0, 1, 2, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 2, 3, 4, 1
Offset: 1
For n=18, we get b^262144 + 1 is prime for b=24518, 40734, 145310, 361658, 525094, ...; the first 3 of these b values are strictly below 262144, hence a(18)=3.
The corresponding primes are 2^4+1; 2^8+1, 4^8+1; 2^16+1; 30^32+1; 120^128+1; 46^512+1; etc.
Cf.
A056993,
A121270,
A259835,
A005574,
A000068,
A006314,
A006313,
A006315,
A006316,
A056994,
A056995,
A057465,
A057002,
A088361,
A088362,
A226528,
A226529,
A226530,
A251597,
A253854,
A244150,
A243959,
A321323, etc.
-
Table[Count[Range[2, 2^n - 1, 2], b_ /; PrimeQ[b^(2^n) + 1]], {n, 9}] (* Michael De Vlieger, Nov 10 2016 *)
-
a(n)=sum(k=1,2^(n-1)-1,ispseudoprime((2*k)^2^n+1)) \\ slow, only probabilistic primality test
Comments