A046052 Number of prime factors of Fermat number F(n).

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 5

Offset: 0

Author

Eric W. Weisstein

Keywords

• nonn
• more
• hard

Comments

F(12) has 6 known factors with C1133 remaining. [Updated by Walter Nissen, Apr 02 2010]

F(13) has 4 known factors with C2391 remaining.

F(14) has one known factor with C4880 remaining. [Updated by Matt C. Anderson, Feb 14 2010]

John Selfridge apparently conjectured that this sequence is not monotonic, so at some point a(n+1) < a(n). Related sequences such as A275377 and A275379 already exhibit such behavior. - Jeppe Stig Nielsen, Jun 08 2018

Factors are counted with multiplicity although it is unknown if all Fermat numbers are squarefree. - Jeppe Stig Nielsen, Jun 09 2018

Links

• W. Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
• W. Keller, Summary of factoring status for Fermat numbersF(n)
• PSI (The algorithm company), Fermat factor status [Broken link?]
• Lorenzo Sauras-Altuzarra, Some properties of the factors of Fermat numbers, Art Discrete Appl. Math. (2022).
• Eric Weisstein's World of Mathematics, Fermat Number
• Wikipedia, Selfridge's Conjecture about Fermat Numbers

Crossrefs

Cf. A000215, A023394, A229850.

Programs

Mathematica

Array[PrimeOmega[2^(2^#) + 1] &, 9, 0] (* Michael De Vlieger, May 31 2022 *)

PARI

a(n)=bigomega(2^(2^n)+1) \\ Eric Chen, Jun 13 2018

Formula

a(n) = A001222(A000215(n)).

Extensions

Name corrected by Arkadiusz Wesolowski, Oct 31 2011