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

1,1

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

Cf. A000215, A204620, A291050.

Sequence in context: A093183 A278380 A290774 * A265956 A189805 A230145

Adjacent sequences: A229849 A229850 A229851 * A229853 A229854 A229855

Keyword

nonn,easy

Author

Arkadiusz Wesolowski, Oct 01 2013

Status

approved