A229850 -id:A229850 - 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

A229854 Primes of the form 384*k + 1.

Original entry on oeis.org

769, 1153, 2689, 3457, 4993, 6529, 7297, 7681, 9601, 10369, 10753, 12289, 13441, 14593, 15361, 18049, 18433, 20353, 21121, 22273, 23041, 26113, 26497, 26881, 29569, 31489, 31873, 32257, 33409, 36097, 37633, 39937, 43777, 45697, 49537, 49921, 52609, 53377

Offset: 1

Arkadiusz Wesolowski, Oct 01 2013

nonn

Not every composite Fermat number has a prime factor of the form 384*k + 1.

Wikipedia, Fermat number

Subsequence of A002476 (primes of form 6m + 1).

Cf. A000215, A023394, A094358, A229850, A229853, A229855, A229856.

[384*n+1 : n in [1..139] | IsPrime(384*n+1)];

Select[Table[384*n + 1, {n, 139}], PrimeQ]

A351332 Primes congruent to 1 (mod 3) that divide some Fermat number.

Original entry on oeis.org

274177, 319489, 6700417, 825753601, 1214251009, 6487031809, 646730219521, 6597069766657, 25409026523137, 31065037602817, 46179488366593, 151413703311361, 231292694251438081, 1529992420282859521, 2170072644496392193, 3603109844542291969

Offset: 1

Arkadiusz Wesolowski, Feb 07 2022

nonn

Subsequence of A014752.

			a(1) = 503^2 + 27*28^2 = 274177 is a prime factor of 2^(2^6) + 1;
a(2) = 383^2 + 27*80^2 = 319489 is a prime factor of 2^(2^11) + 1;
a(3) = 887^2 + 27*468^2 = 6700417 is a prime factor of 2^(2^5) + 1;
a(4) = 27017^2 + 27*1884^2 = 825753601 is a prime factor of 2^(2^16) + 1;
a(5) = 2561^2 + 27*6688^2 = 1214251009 is a prime factor of 2^(2^15) + 1;

Allan Cunningham, Haupt-exponents of 2, The Quarterly Journal of Pure and Applied Mathematics, Vol. 37 (1906), pp. 122-145.

Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m.

Cf. A002476, A014752, A023394, A204620, A229850, A229853.

isok(p) = if(p%6==1 && isprime(p), my(z=znorder(Mod(2, p))); z>>valuation(z, 2)==1, return(0));

A002476 INTERSECT A023394.

A360652 Primes of the form x^2 + 432*y^2.

Original entry on oeis.org

433, 457, 601, 1657, 1753, 1777, 1801, 2017, 2089, 2113, 2281, 2689, 2833, 2953, 3457, 3889, 4057, 4129, 4153, 4177, 4513, 4657, 4729, 5113, 5209, 5449, 5569, 5737, 5953, 6217, 6361, 6673, 6961, 7057, 7321, 7369, 7537, 7753, 7873, 8353, 8377, 8713, 8761, 8929

Offset: 1

Arkadiusz Wesolowski, Feb 15 2023

nonn

Supersequence of A351332. Thus every prime congruent to 1 mod 3 that divides a Fermat number is in this sequence.

Every Fermat number that is a semiprime has a prime of this form as a factor.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Cf. A014752, A023394, A229850, A351332.

[p: p in PrimesUpTo(8929) | NormEquation(432, p) eq true];

select(p->my(m=Mod(2, p)^(p\12)); p>11 && (m==1||m==p-1), primes(1110))

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A229854 Primes of the form 384*k + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

A351332 Primes congruent to 1 (mod 3) that divide some Fermat number.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Formula

A360652 Primes of the form x^2 + 432*y^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

PARI