A366671 Smallest prime dividing 8^n + 1.

2, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 193, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 641, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 193, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 769, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5

Offset: 0

Author

Sean A. Irvine, Oct 15 2023

Keywords

• nonn

Comments

a(n) = 3 if n is odd. a(n) = 5 if n == 2 (mod 4). - Robert Israel, Nov 20 2023

Links

• Max Alekseyev, Table of n, a(n) for n = 0..16383

Crossrefs

Cf. A002586, A366609, A052536, A366655, A274905, A366670, A038371, A366719.

Programs

Maple

P1000:= mul(ithprime(i),i= 4..1000):
f:= proc(n) local t;
  if n::odd then return 3 elif n mod 4 = 2 then return 5 fi;
  t:= igcd(8^n+1,P1000);
  if t <> 1 then min(numtheory:-factorset(t)) else min(numtheory:-factorset(8^n+1)) fi
end proc:
map(f, [$0..100]); # Robert Israel, Nov 20 2023

Mathematica

Table[FactorInteger[8^n + 1][[1,1]], {n, 0, 78}] (* Paul F. Marrero Romero, Oct 20 2023 *)

Python

from sympy import primefactors
def A366671(n): return min(primefactors((1<<3*n)+1)) # Chai Wah Wu, Oct 16 2023

Formula

a(n) = A020639(A062395(n)). - Paul F. Marrero Romero, Oct 20 2023

a(n) = A002586(3*n) for n >= 1. - Robert Israel, Nov 20 2023