A007117 - OEIS

%I M4586 #34 Mar 02 2021 09:22:15

%S 0,0,1,8,1024,5,1071,116503103764643,1209889024954,1184,11131,39,7,

%T 82731770,1784180997819127957596374417642156545110881094717,9264,3150,

%U 59251857,13,33629

%N a(0) = a(1) = 0; for n >= 2, a(n)*2^(n+2) + 1 is the 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

%D P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 71.

%D H. Riesel, ``Prime numbers and computer methods for factorization,'' Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see p. 377.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Wilfrid Keller, <a href="http://www.prothsearch.com/fermat.html">Prime factors k.2^n + 1 of Fermat numbers F_m</a>

%H R. G. Wilson, V, <a href="/A007117/a007117.pdf">Letter to N. J. A. Sloane, Aug. 1993</a>

%F a(n) = (A093179(n) - 1)/2^(n+2) for n >= 2. - _Jianing Song_, Mar 02 2021

%e From _Jianing Song_, Mar 02 2021: (Start)

%e F(2) = 2^(2^2) + 1 = 1*2^4 + 1;

%e F(3) = 2^(2^3) + 1 = 5*2^5 + 1;

%e F(4) = 2^(2^4) + 1 = 1024*2^6 + 1;

%e F(5) = 2^(2^5) + 1 = (5*2^7 + 1) * (52347*2^7 + 1);

%e F(6) = 2^(2^6) + 1 = (1071*2^8 + 1) * (262814145745*2^8 + 1). (End)

%o (PARI) a(n) = if(n<2, 0, my(lim=2^(2^n-(n+2))); for(k=1, lim, my(p=k*2^(n+2)+1); if(Mod(2,p)^(2^n)==-1, return(k)))) \\ _Jianing Song_, Mar 02 2021

%Y Cf. A093179.

%K hard,nonn

%O 0,4

%A _N. J. A. Sloane_, _Robert G. Wilson v_

%E a(14)-a(19) added by _Max Alekseyev_, May 04 2010