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A229852 3*h^2, where h is an odd integer not divisible by 3. 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 (list; graph; refs; listen; history; text; internal format)
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
Wilfrid Keller, Fermat factoring status.
Eric Weisstein's World of Mathematics, Fermat Number.
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
Sequence in context: A093183 A278380 A290774 * A265956 A189805 A230145
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified June 16 08:45 EDT 2024. Contains 373424 sequences. (Running on oeis4.)