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 A064386 Numbers of the form 2^k+1 or 4^k-2^k+1. 2
 1, 2, 3, 5, 9, 13, 17, 33, 57, 65, 129, 241, 257, 513, 993, 1025, 2049, 4033, 4097, 8193, 16257, 16385, 32769, 65281, 65537, 131073, 261633, 262145, 524289, 1047553, 1048577, 2097153, 4192257, 4194305, 8388609, 16773121, 16777217 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. REFERENCES J. F. Dillon, Geometry, codes and difference sets: exceptional connections, in Codes and designs (Columbus, OH, 2000), pp. 73-85, de Gruyter, Berlin, 2002. LINKS Robert Israel, Table of n, a(n) for n = 1..4978 H. Janwa, G. McGuire and R. M. Wilson, Double-error-correcting codes and absolutely irreducible polynomials over GF(2), J. Algebra, 178 (1995), 665-676. FORMULA Conjectures from Colin Barker, Mar 14 2018: (Start) 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)). a(n) = a(n-1) + 6*a(n-3) - 6*a(n-4) - 8*a(n-6) + 8*a(n-7) for n>7. (End) MAPLE N:= 10^11: # to get all terms <= N sort([1, seq(2^n+1, n=0..ilog2(N-1)), seq(4^n-2^n+1, n=2..floor(log((sqrt(4*N-3)+1)/2)))]); # Robert Israel, Mar 14 2018 MATHEMATICA 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 *) CROSSREFS Cf. A064390. Sequence in context: A221708 A256408 A256404 * A049715 A076095 A085913 Adjacent sequences:  A064383 A064384 A064385 * A064387 A064388 A064389 KEYWORD nonn AUTHOR N. J. A. Sloane, Sep 28 2001 STATUS approved

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Last modified June 19 15:58 EDT 2021. Contains 345144 sequences. (Running on oeis4.)