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a(n) = A091255(n, n + 1).
3

%I #11 Jan 21 2024 09:37:40

%S 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,

%T 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,

%U 1,1,3,1,1,1,1,1,3,1,1,1,1,1,7,1,3,1,1

%N a(n) = A091255(n, n + 1).

%C 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.

%H <a href="/index/Ge#GF2X">Index entries for sequences operating on GF(2)[X]-polynomials</a>

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

%e The first terms, alongside the correspond GF(2)[X]-polynomials, are:

%e n a(n) P(n) P(n+1) gcd(P(n), P(n+1))

%e -- ---- ----------------- ----------------- -----------------

%e 1 1 1 X 1

%e 2 1 X X + 1 1

%e 3 1 X + 1 X^2 1

%e 4 1 X^2 X^2 + 1 1

%e 5 3 X^2 + 1 X^2 + X X + 1

%e 6 1 X^2 + X X^2 + X + 1 1

%e 7 1 X^2 + X + 1 X^3 1

%e 8 1 X^3 X^3 + 1 1

%e 9 3 X^3 + 1 X^3 + X X + 1

%e 10 1 X^3 + X X^3 + X + 1 1

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

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

%K nonn,base

%O 1,5

%A _Rémy Sigrist_, Jan 19 2024