

A093179


Smallest factor of the nth Fermat number F(n) = 2^(2^n) + 1.


10



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

0,1


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


LINKS

Table of n, a(n) for n=0..19.
Ivars Peterson, Cracking Fermat Numbers
Wilfrid Keller, Prime factors k·2^n + 1 of Fermat numbers F_m and complete factoring status
Eric Weisstein's World of Mathematics, Fermat Number


FORMULA

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


EXAMPLE

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.


MATHEMATICA

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


PROG

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


CROSSREFS

Cf. A000051, A070592, A007117.
Leading entries in triangle A050922.
Sequence in context: A019434 A164307 A125045 * A067387 A050922 A260476
Adjacent sequences: A093176 A093177 A093178 * A093180 A093181 A093182


KEYWORD

nonn,hard


AUTHOR

Eric W. Weisstein, Mar 27 2004


EXTENSIONS

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar
Added a(14)a(15), Jeppe Stig Nielsen, Feb 11 2010
a(16)a(19) added based on terms of A007117 by Jianing Song, Mar 02 2021


STATUS

approved



