A093179 Smallest prime factor of the n-th Fermat number F(n) = 2^(2^n) + 1.

3, 5, 17, 257, 65537, 641, 274177, 59649589127497217, 1238926361552897, 2424833, 45592577, 319489, 114689, 2710954639361, 116928085873074369829035993834596371340386703423373313, 1214251009, 825753601, 31065037602817, 13631489, 70525124609

Offset: 0

Author

Eric W. Weisstein, Mar 27 2004

Keywords

• nonn
• hard

Comments

a(14) might need to be corrected if F(14) turns out to have a smaller factor than 116928085873074369829035993834596371340386703423373313. F(20) is composite, but no explicit factor is known. - Jeppe Stig Nielsen, Feb 11 2010

Examples

			F(0) = 2^(2^0) + 1 = 3, prime.
F(5) = 2^(2^5) + 1 = 4294967297 = 641*6700417.
So 3 as the 0th entry and 641 is the 5th term.
		

References

• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 73.

Links

• Wilfrid Keller, Prime factors k·2^n + 1 of Fermat numbers F_m and complete factoring status
• Ivars Peterson, Cracking Fermat Numbers
• Lorenzo Sauras-Altuzarra, Some properties of the factors of Fermat numbers, Art Discrete Appl. Math. (2022).
• Eric Weisstein's World of Mathematics, Fermat Number

Crossrefs

Cf. A000051, A000215, A020639, A070592, A007117.

Leading entries in triangle A050922.

Programs

Mathematica

Table[With[{k = 2^n}, FactorInteger[2^k + 1]][[1, 1]], {n, 0, 15, 1}] (* Vincenzo Librandi, Jul 23 2013 *)

PARI

g(n)=for(x=9,n,y=Vec(ifactor(2^(2^x)+1));print1(y[1]",")) \\ Cino Hilliard, Jul 04 2007

Formula

a(n) = A007117(n)*2^(n+2) + 1 for n >= 2. - Jianing Song, Mar 02 2021

a(n) = A020639(A000215(n)). - Alois P. Heinz, Oct 25 2024

Extensions

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar

a(14)-a(15) added by Jeppe Stig Nielsen, Feb 11 2010

a(16)-a(19) added based on terms of A007117 by Jianing Song, Mar 02 2021