A064386 Numbers of the form 2^k+1 or 4^k-2^k+1.

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

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[2]((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