A000068 -id:A000068 - cp's OEIS frontend

A272137 Primes of the form k^16 + 1.

2, 65537, 197352587024076973231046657, 808551180810136214718004658177, 1238846438084943599707227160577, 37157429083410091685945089785857, 123025056645280288014028950372089857, 150838912030874130174020868290707457

Offset: 1

Jaroslav Krizek, May 08 2016

nonn

Corresponding values of k are in A006313.

Jaroslav Krizek, Table of n, a(n) for n = 1..100

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).

Corresponding sequences of primes p of the form n^(2^k)+1 for k = 1-4: A002496 (k=1), A037896 (k=2), A258805 (k=3), A272137 (k=4).

[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

A135047 Initial members of an octuplet of generalized Fermat primes: numbers n such that (n+m)^4+1 is prime for m=0,2,4,6,8,10,12 and 14.

Original entry on oeis.org

10332305196, 15731023654, 202193785336, 417860702688, 427241399860, 648488931216

Offset: 1

Martin Raab, Feb 11 2008

nonn

n^4+1 can be prime for at most eight consecutive even numbers n, otherwise at least one member would be divisible by 17.

			a(1)=10332305196 because 103323051964^4+1 is prime and (10332305196+m)^4+1 is prime for all even m up to 14.

Cf. A000068.

A242552 Least number k such that n^4 + k^4 is prime.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 3, 2, 7, 2, 13, 4, 5, 8, 1, 2, 5, 2, 1, 10, 15, 2, 1, 6, 3, 2, 1, 12, 7, 12, 5, 14, 1, 6, 7, 2, 3, 14, 9, 2, 5, 10, 21, 2, 1, 4, 1, 2, 7, 2, 11, 6, 1, 14, 1, 2, 7, 2, 11, 2, 11, 8, 23, 16, 29, 12, 3, 10, 27, 2, 5, 8, 1, 8, 3, 20, 17, 2, 1, 10, 1, 10

Offset: 1

Derek Orr, May 17 2014

nonn

If a(n) = 1, then n is in A000068.

			8^4+1^4 = 4097 is not prime. 8^4+2^4 = 4112 is not prime. 8^4+3^4 = 4177 is prime. Thus, a(8) = 3.

Cf. A069003, A000068.

a(n)=for(k=1,oo,if(ispseudoprime(n^4+k^4),return(k)));

import sympy
from sympy import isprime
def a(n):
  for k in range(10**4):
    if isprime(n**4+k**4):
      return k
n = 1
while n < 100:
  print(a(n))
  n += 1

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

cp's OEIS Frontend

A272137 Primes of the form k^16 + 1.

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A135047 Initial members of an octuplet of generalized Fermat primes: numbers n such that (n+m)^4+1 is prime for m=0,2,4,6,8,10,12 and 14.

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A242552 Least number k such that n^4 + k^4 is 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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