|
%I #40 Jan 31 2023 16:36:13
%S 3,5,17,257,65537,641,274177,59649589127497217,1238926361552897,
%T 2424833,45592577,319489,114689,2710954639361,
%U 116928085873074369829035993834596371340386703423373313,1214251009,825753601,31065037602817,13631489,70525124609
%N Smallest prime factor of the n-th Fermat number F(n) = 2^(2^n) + 1.
%C 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
%H Wilfrid Keller, <a href="http://www.prothsearch.com/fermat.html">Prime factors k·2^n + 1 of Fermat numbers F_m and complete factoring status</a>
%H Ivars Peterson, <a href="https://www.sciencenews.org/article/cracking-fermat-numbers">Cracking Fermat Numbers</a>
%H Lorenzo Sauras-Altuzarra, <a href="https://doi.org/10.26493/2590-9770.1473.ec5">Some properties of the factors of Fermat numbers</a>, Art Discrete Appl. Math. (2022).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FermatNumber.html">Fermat Number</a>
%F a(n) = A007117(n)*2^(n+2) + 1 for n >= 2. - _Jianing Song_, Mar 02 2021
%e F(0) = 2^(2^0) + 1 = 3, prime.
%e F(5) = 2^(2^5) + 1 = 4294967297 = 641*6700417.
%e So 3 as the 0th entry and 641 is the 5th term.
%t Table[With[{k = 2^n}, FactorInteger[2^k + 1]][[1, 1]], {n, 0, 15, 1}] (* _Vincenzo Librandi_, Jul 23 2013 *)
%o (PARI) g(n)=for(x=9,n,y=Vec(ifactor(2^(2^x)+1));print1(y[1]",")) \\ _Cino Hilliard_, Jul 04 2007
%Y Cf. A000051, A070592, A007117.
%Y Leading entries in triangle A050922.
%K nonn,hard
%O 0,1
%A _Eric W. Weisstein_, Mar 27 2004
%E Edited by _N. J. A. Sloane_, Jul 02 2008 at the suggestion of _R. J. Mathar_
%E a(14)-a(15) added by _Jeppe Stig Nielsen_, Feb 11 2010
%E a(16)-a(19) added based on terms of A007117 by _Jianing Song_, Mar 02 2021
| 