A023395 Only Fermat number divisible by A023394(n) is 2^2^a(n) + 1.

0, 1, 2, 3, 5, 4, 12, 6, 11, 11, 9, 5, 18, 12, 10, 12, 23, 16, 15, 10, 19, 12, 19, 13, 36, 21, 38, 32, 25, 17, 39, 6, 26, 27, 30, 30, 8, 12, 15, 29, 38, 7, 25, 27, 36, 42, 25, 13, 13, 55

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• more

Comments

From Jianing Song, Mar 02 2021: (Start)

2^(a(n)+1) is the multiplicative order of 2 modulo A023394(n).

Each k occurs A046052(k) times in this sequence provided that F(k) = 2^2^k + 1 is squarefree (no counterexamples are known). (End)

Alternatively, a(n) is the only k such that A023394(n) divides A000215(k). - Lorenzo Sauras Altuzarra, Feb 01 2023

Links

• Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
• Lorenzo Sauras-Altuzarra, Some properties of the factors of Fermat numbers, Art Discrete Appl. Math. (2022).
• Index entries for sequences that are related to primes dividing Fermat numbers

Crossrefs

Cf. A000215, A023394, A046052.

Programs

PARI

forprime(p=3,,r=znorder(Mod(2,p));hammingweight(r)==1&&print1(logint(r,2)-1,", ")) \\ Jeppe Stig Nielsen, Mar 04 2018

Extensions

a(25)-a(41) computed using data from Wilfrid Keller by T. D. Noe, Feb 01 2009

Three more terms by T. D. Noe, Feb 03 2009

Six more terms from Wilfrid Keller by T. D. Noe, Jan 14 2013