A229852 3*h^2, where h is an odd integer not divisible by 3.

3, 75, 147, 363, 507, 867, 1083, 1587, 1875, 2523, 2883, 3675, 4107, 5043, 5547, 6627, 7203, 8427, 9075, 10443, 11163, 12675, 13467, 15123, 15987, 17787, 18723, 20667, 21675, 23763, 24843, 27075, 28227, 30603, 31827, 34347, 35643, 38307, 39675, 42483, 43923

Offset: 1

Author

Arkadiusz Wesolowski, Oct 01 2013

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000215, A204620, A291050.

Programs

Magma

[3*h^2 : h in [1..121 by 2] | not IsZero(h mod 3)];

Mathematica

3*Select[Range[1, 121, 2], Mod[#, 3] > 0 &]^2 (* Amiram Eldar, Jan 02 2021 *)

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

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