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A270271 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}.

%I A270271 #15 Apr 03 2023 10:36:13
%S A270271 201446503145165177,1007236913771681629,1697906240793858917,
%T A270271 2331023822106839599,2935363331541925531,3367034409844073483,
%U A270271 3914042604075779837,4863495246870308311,5036162578625852633,5590196669446332863,6705290764721718679,7284449444083822547
%N A270271 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 A270271 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 A270271 Arkadiusz Wesolowski, <a href="/A270271/b270271.txt">Table of n, a(n) for n = 1..64</a>
%H A270271 Chris Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/xpage/SierpinskiNumber.html">Sierpinski number</a>
%H A270271 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 A270271 <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 A270271 a(n) = a(n-64) + 2*(2^64-1) for n > 64.
%o A270271 (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 A270271 Cf. A257647. Subsequence of A076336.
%K A270271 nonn
%O A270271 1,1
%A A270271 _Arkadiusz Wesolowski_, Mar 14 2016