A007117 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.

0, 0, 1, 8, 1024, 5, 1071, 116503103764643, 1209889024954, 1184, 11131, 39, 7, 82731770, 1784180997819127957596374417642156545110881094717, 9264, 3150, 59251857, 13, 33629

Offset: 0

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

• hard
• nonn

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

			From _Jianing Song_, Mar 02 2021: (Start)
F(2) = 2^(2^2) + 1 = 1*2^4 + 1;
F(3) = 2^(2^3) + 1 = 5*2^5 + 1;
F(4) = 2^(2^4) + 1 = 1024*2^6 + 1;
F(5) = 2^(2^5) + 1 = (5*2^7 + 1) * (52347*2^7 + 1);
F(6) = 2^(2^6) + 1 = (1071*2^8 + 1) * (262814145745*2^8 + 1). (End)
		

References

• P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 71.
• H. Riesel, ``Prime numbers and computer methods for factorization,'' Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see p. 377.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
• R. G. Wilson, V, Letter to N. J. A. Sloane, Aug. 1993

Crossrefs

Cf. A093179.

Programs

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

Formula

a(n) = (A093179(n) - 1)/2^(n+2) for n >= 2. - Jianing Song, Mar 02 2021

Extensions

a(14)-a(19) added by Max Alekseyev, May 04 2010