A369318 Numbers k such that A091255(k, k + 1) <> 1.

5, 9, 17, 23, 27, 29, 33, 35, 39, 45, 53, 57, 65, 71, 77, 83, 85, 89, 95, 101, 105, 107, 113, 119, 125, 129, 135, 139, 141, 149, 153, 159, 165, 169, 177, 179, 183, 189, 195, 197, 201, 209, 215, 221, 223, 225, 231, 237, 245, 249, 251, 257, 259, 263, 269, 277

Offset: 1

Author

Rémy Sigrist, Jan 19 2024

Keywords

• nonn
• base

Comments

Equivalently, numbers k such that A369317(k) <> 1.

Two consecutive integers are always coprime, however the polynomials over GF(2) whose coefficients are encoded in the binary expansions of two consecutive integers are not necessarily coprime, hence this sequence.

Examples

			The first terms, alongside the correspond GF(2)[X]-polynomials, are:
  n   a(n)  P(a(n))              P(a(n)+1)            gcd(P(a(n)), P(a(n)+1))
  --  ----  -------------------  -------------------  -----------------------
   1     5  X^2 + 1              X^2 + X              X + 1
   2     9  X^3 + 1              X^3 + X              X + 1
   3    17  X^4 + 1              X^4 + X              X + 1
   4    23  X^4 + X^2 + X + 1    X^4 + X^3            X + 1
   5    27  X^4 + X^3 + X + 1    X^4 + X^3 + X^2      X^2 + X + 1
   6    29  X^4 + X^3 + X^2 + 1  X^4 + X^3 + X^2 + X  X + 1
   7    33  X^5 + 1              X^5 + X              X + 1
   8    35  X^5 + X + 1          X^5 + X^2            X^2 + X + 1
   9    39  X^5 + X^2 + X + 1    X^5 + X^3            X^2 + 1
  10    45  X^5 + X^3 + X^2 + 1  X^5 + X^3 + X^2 + X  X + 1
		

Links

• Index entries for sequences operating on GF(2)[X]-polynomials

Crossrefs

Cf. A091255, A369317.

Programs

PARI

is(n) = fromdigits(lift(Vec(gcd(Mod(1, 2) * Pol(binary(n)), Mod(1, 2) * Pol(binary(n+1))))), 2) != 1