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%I #17 Jul 26 2019 17:08:48
%S 1,2,3,5,9,13,17,33,57,65,129,241,257,513,993,1025,2049,4033,4097,
%T 8193,16257,16385,32769,65281,65537,131073,261633,262145,524289,
%U 1047553,1048577,2097153,4192257,4194305,8388609,16773121,16777217
%N Numbers of the form 2^k+1 or 4^k-2^k+1.
%C Call m exceptional if the binary cyclic code of length 2^k-1 with zeros w and w^m (w primitive in GF(2^k)) is double-error-correcting for infinitely many k. It is conjectured that this sequence (with the initial terms 1 and 2 omitted) gives all odd exceptional m's.
%D J. F. Dillon, Geometry, codes and difference sets: exceptional connections, in Codes and designs (Columbus, OH, 2000), pp. 73-85, de Gruyter, Berlin, 2002.
%H Robert Israel, <a href="/A064386/b064386.txt">Table of n, a(n) for n = 1..4978</a>
%H H. Janwa, G. McGuire and R. M. Wilson, <a href="https://doi.org/10.1006/jabr.1995.1372">Double-error-correcting codes and absolutely irreducible polynomials over GF(2)</a>, J. Algebra, 178 (1995), 665-676.
%F Conjectures from _Colin Barker_, Mar 14 2018: (Start)
%F G.f.: x*(1 + x + x^2 - 4*x^3 - 2*x^4 - 2*x^5 + 8*x^8) / ((1 - x)*(1 - 2*x^3)*(1 - 4*x^3)).
%F a(n) = a(n-1) + 6*a(n-3) - 6*a(n-4) - 8*a(n-6) + 8*a(n-7) for n>7.
%F (End)
%p N:= 10^11: # to get all terms <= N
%p sort([1,seq(2^n+1, n=0..ilog2(N-1)), seq(4^n-2^n+1, n=2..floor(log[2]((sqrt(4*N-3)+1)/2)))]); # _Robert Israel_, Mar 14 2018
%t With[{nn=40},Take[Flatten[Table[{2^n+1,4^n-2^n+1},{n,0,nn}]]//Union,40]] (* _Harvey P. Dale_, Jul 26 2019 *)
%Y Cf. A064390.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Sep 28 2001
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