A272486 Numbers n > 1 such that the polynomial x^(n-k)*(x+1)^k+1 over GF(2) is reducible for all k from 1 to n-1.

8, 13, 16, 19, 24, 26, 27, 32, 37, 38, 40, 42, 43, 45, 48, 50, 51, 53, 54, 56, 59, 61, 64, 66, 67, 69, 70, 72, 75, 77, 78, 80, 82, 83, 85, 88, 90, 91, 96, 99, 101, 104, 107, 109, 110, 112, 114, 115, 116, 117, 120, 122, 125, 126, 128, 131, 133, 136, 138, 139

Offset: 1

Author

Luis H. Gallardo, May 01 2016

Keywords

• nonn

Comments

If n = p is an odd prime, the sequence appears coincide with A057749. Is this true, and if so, why?

More generally, the sequence appears the same as the set complement of A073571 (minus {1}). Again, if true, why? [This is not correct. E.g., the terms 42, 54, 66, 90, and 110 are in both sequences. Joerg Arndt, Apr 29 2019]

Crossrefs

Cf. A057749, A073571.

Programs

Mathematica

ok[n_] := {} == Quiet@ Select[Range[n-1], IrreduciblePolynomialQ[ x^(n-#) * (x+1)^# + 1, Modulus -> 2] &, 1]; Select[Range[2, 140], ok] (* Giovanni Resta, May 02 2016 *)

PARI

isok(n) = if (n<=1, 0, for (k=1, n-1, if (polisirreducible(Mod(1,2)*(x^(n-k)*(x+1)^k+1)), return(0));); 1;); \\ Michel Marcus, May 02 2016

Extensions

a(41)-a(60) from Giovanni Resta, May 02 2016