A275377 -id:A275377 - cp's OEIS frontend

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

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

Offset: 0

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

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

Cf. A000215, A023394, A229850.

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

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

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

Name corrected by Arkadiusz Wesolowski, Oct 31 2011

A302097 Number of odd prime factors (with multiplicity) of generalized Fermat number 13^(2^n) + 1.

Original entry on oeis.org

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

Offset: 0

Jeppe Stig Nielsen, Apr 01 2018

a(8) >= 6. - Chai Wah Wu, Dec 09 2019

			b(n) = (13^(2^n) + 1)/2.
Complete factorizations:
b(0) = 7
b(1) = 5*17
b(2) = 14281
b(3) = 407865361
b(4) = 2657*441281*283763713
b(5) = 193*1601*10433*68675120456139881482562689
b(6) = 257*3230593*36713826768408543617*3215877717636198473712500018174097551256193
b(7) = 96769*2940673*P131

factordb.com query on (13^256+1)/2.

Cf. A275377, A275378, A275379, A275380, A275381, A275382, A275383, A302098.

PARI
```
a(n) = bigomega((13^(2^n)+1)/2)
```

A302098 Number of prime factors (with multiplicity) of generalized Fermat number 14^(2^n) + 1.

Original entry on oeis.org

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

Offset: 0

Jeppe Stig Nielsen, Apr 01 2018

a(8) >= 5. - Chai Wah Wu, Dec 09 2019

			b(n) = 14^(2^n) + 1
Complete factorizations:
b(0) = 3*5
b(1) = 197
b(2) = 41*937
b(3) = 17*5393*16097
b(4) = 193*11284732320255809
b(5) = 7489*1204905857*1667461121*315256811699009
b(6) = 8633886977*P64
b(7) = 257*100497382788383295179961898289105815085380571534081*P95

factordb.com query on 14^256+1.

Cf. A275377, A275378, A275379, A275380, A275381, A275382, A275383, A302097, A152587.

PARI
```
a(n) = bigomega(14^(2^n)+1)
```

a(n) = A001222(A152587(n)).

cp's OEIS Frontend

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

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A302097 Number of odd prime factors (with multiplicity) of generalized Fermat number 13^(2^n) + 1.

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A302098 Number of prime factors (with multiplicity) of generalized Fermat number 14^(2^n) + 1.

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