A006313 -id:A006313 - cp's OEIS frontend

A217991 Numbers n such that n^16+1 and (n+2)^16+1 are both prime.

74, 156, 474, 476, 686, 688, 774, 776, 830, 832, 982, 2698, 3560, 3646, 4016, 5974, 8542, 8732, 10730, 10858, 12164, 12902, 13022, 13154, 13554, 13838, 13840, 20436, 20454, 20504, 21912, 24554, 24770, 25582, 30144, 31634, 31882, 32358, 32360, 32992, 33758

Offset: 1

Michel Lagneau, Oct 17 2012

			74 is in the sequence because 74^16+1 =  808551180810136214718004658177 and 76^16+1 = 1238846438084943599707227160577 are both prime.

Cf. A006313.

lst={}; Do[p=n^16+1; q=(n+2)^16+1;If[PrimeQ[p] && PrimeQ[q], AppendTo[lst, n]], {n, 0, 34000}];lst
Select[Range[34000],AllTrue[{#^16+1,(#+2)^16+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 24 2015 *)

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

Jeppe Stig Nielsen, Nov 06 2016

The choice whether to take b < 2^n or b <= 2^n matters only for n=1 and n=2 unless there are more primes like 2^2+1 and 4^4+1 (see A121270).
Perfect squares b are allowed.
a(20) was determined after a lengthy computation by distributed project PrimeGrid, cf. A321323. - Jeppe Stig Nielsen, Jan 02 2019

			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.

Y. Gallot, Status of the smallest base values yielding Generalized Fermat primes.
PrimeGrid, GFN Prime Search Status and History.

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

a(20) from Jeppe Stig Nielsen, Jan 02 2019

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A217991 Numbers n such that n^16+1 and (n+2)^16+1 are both prime.

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A277967 Number of even numbers b with 0 < b < 2^n such that b^(2^n) + 1 is prime.

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