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

%I A229852 #33 Jul 12 2025 21:31:45
%S A229852 3,75,147,363,507,867,1083,1587,1875,2523,2883,3675,4107,5043,5547,
%T A229852 6627,7203,8427,9075,10443,11163,12675,13467,15123,15987,17787,18723,
%U A229852 20667,21675,23763,24843,27075,28227,30603,31827,34347,35643,38307,39675,42483,43923
%N A229852 3*h^2, where h is an odd integer not divisible by 3.
%C A229852 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.
%C A229852 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).
%D A229852 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.
%H A229852 Colin Barker, <a href="/A229852/b229852.txt">Table of n, a(n) for n = 1..1000</a>
%H A229852 Wilfrid Keller, <a href="http://www.prothsearch.com/fermat.html">Fermat factoring status</a>.
%H A229852 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FermatNumber.html">Fermat Number</a>.
%H A229852 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A229852 G.f.: 3*x*(1+24*x+22*x^2+24*x^3+x^4) / ((1-x)^3*(1+x)^2).
%F A229852 a(n) = 3*A104777(n).
%F A229852 From _Colin Barker_, Jan 26 2016: (Start)
%F A229852 a(n) = 3*(18*n^2+6*(-1)^n*n-18*n-3*(-1)^n+5)/2.
%F A229852 a(n) = 27*n^2-18*n+3 for n even.
%F A229852 a(n) = 27*n^2-36*n+12 for n odd.
%F A229852 (End)
%F A229852 Sum_{n>=1} 1/a(n) = Pi^2/27 (A291050). - _Amiram Eldar_, Jan 02 2021
%t A229852 3*Select[Range[1, 121, 2], Mod[#, 3] > 0 &]^2 (* _Amiram Eldar_, Jan 02 2021 *)
%o A229852 (Magma) [3*h^2 : h in [1..121 by 2] | not IsZero(h mod 3)];
%o A229852 (PARI) forstep(h=1, 121, 2, if(!(h%3==0), print1(3*h^2, ", ")));
%o A229852 (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
%Y A229852 Cf. A000215, A204620, A291050.
%K A229852 nonn,easy
%O A229852 1,1
%A A229852 _Arkadiusz Wesolowski_, Oct 01 2013