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. (Formerly M4586)

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

Offset

0,4

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

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

Table of n, a(n) for n=0..19.

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

Formula

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

Example

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)

Prog

(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

Crossrefs

Cf. A093179.

Sequence in context: A260028 A176367 A298669 * A291831 A085533 A360773

Adjacent sequences: A007114 A007115 A007116 * A007118 A007119 A007120

Keyword

hard,nonn

Author

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

Extensions

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

Status

approved