OFFSET
1,1
COMMENTS
If p = a(n)*2^k + 1 divides a composite Fermat number 2^(2^m) + 1 and p is a prime, then k is odd.
More precisely, k == 1 (mod 4) if h == +/- 1 (mod 5) and k == 3 (mod 4) if h == +/- 2 (mod 5) (Krizek, Luca and Somer).
REFERENCES
M. Krizek, F. Luca, L. Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS Books in Mathematics, vol. 9, Springer-Verlag, New York, 2001, pp. 63-65.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Wilfrid Keller, Fermat factoring status.
Eric Weisstein's World of Mathematics, Fermat Number.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
FORMULA
G.f.: 3*x*(1+24*x+22*x^2+24*x^3+x^4) / ((1-x)^3*(1+x)^2).
a(n) = 3*A104777(n).
From Colin Barker, Jan 26 2016: (Start)
a(n) = 3*(18*n^2+6*(-1)^n*n-18*n-3*(-1)^n+5)/2.
a(n) = 27*n^2-18*n+3 for n even.
a(n) = 27*n^2-36*n+12 for n odd.
(End)
Sum_{n>=1} 1/a(n) = Pi^2/27 (A291050). - Amiram Eldar, Jan 02 2021
MATHEMATICA
3*Select[Range[1, 121, 2], Mod[#, 3] > 0 &]^2 (* Amiram Eldar, Jan 02 2021 *)
PROG
(Magma) [3*h^2 : h in [1..121 by 2] | not IsZero(h mod 3)]
(PARI) forstep(h=1, 121, 2, if(!(h%3==0), print1(3*h^2, ", ")));
(PARI) Vec(3*x*(1+24*x+22*x^2+24*x^3+x^4) / ((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 26 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Arkadiusz Wesolowski, Oct 01 2013
STATUS
approved