%I #15 Apr 03 2023 10:36:13
%S 201446503145165177,1007236913771681629,1697906240793858917,
%T 2331023822106839599,2935363331541925531,3367034409844073483,
%U 3914042604075779837,4863495246870308311,5036162578625852633,5590196669446332863,6705290764721718679,7284449444083822547
%N Odd numbers n such that for every k >= 1, n*2^k + 1 has a divisor in the set {3, 5, 17, 257, 641, 65537, 6700417}.
%D 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. 72-73.
%H Arkadiusz Wesolowski, <a href="/A270271/b270271.txt">Table of n, a(n) for n = 1..64</a>
%H Chris Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/xpage/SierpinskiNumber.html">Sierpinski number</a>
%H W. Sierpiński, <a href="http://dx.doi.org/10.5169/seals-20713">Sur un problème concernant les nombres k * 2^n + 1</a>, Elem. Math., 15 (1960), pp. 73-74.
%H <a href="/index/Rec#order_65">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).
%F a(n) = a(n-64) + 2*(2^64-1) for n > 64.
%o (Magma) lst:=[]; e:=2^64; P:=PrimeDivisors(e-1); C:=[1, 1, 1, 1, 1, 1, 33]; Pr:=&*[P[i]: i in [1..#P]]; S:=CRT([Modexp(2, C[i], P[i]): i in [1..#C]], P); for t in [1..33] do a:=S+Pr; g:=Gcd(a, e); S:=Floor(a/g); Append(~lst, S); end for; Sort(lst)[1..12];
%Y Cf. A257647. Subsequence of A076336.
%K nonn
%O 1,1
%A _Arkadiusz Wesolowski_, Mar 14 2016