A369317 a(n) = A091255(n, n + 1).

1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 1, 3, 1, 1, 1, 3, 1, 7, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 7, 1, 3, 1, 1

Offset: 1

Author

Rémy Sigrist, Jan 19 2024

Keywords

• nonn
• base

Comments

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(n)               P(n+1)             gcd(P(n), P(n+1))
  --  ----  -----------------  -----------------  -----------------
   1     1  1                  X                  1
   2     1  X                  X + 1              1
   3     1  X + 1              X^2                1
   4     1  X^2                X^2 + 1            1
   5     3  X^2 + 1            X^2 + X            X + 1
   6     1  X^2 + X            X^2 + X + 1        1
   7     1  X^2 + X + 1        X^3                1
   8     1  X^3                X^3 + 1            1
   9     3  X^3 + 1            X^3 + X            X + 1
  10     1  X^3 + X            X^3 + X + 1        1
		

Links

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

Crossrefs

Cf. A091255, A129868, A369277 (distinct values), A369318 (indices of values <> 1).

Programs

PARI

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

Formula

a(A129868(k)) = 2^(k+1) - 1 for any k > 0.