This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A369317 #11 Jan 21 2024 09:37:40 %S A369317 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 A369317 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 A369317 1,1,3,1,1,1,1,1,3,1,1,1,1,1,7,1,3,1,1 %N A369317 a(n) = A091255(n, n + 1). %C A369317 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 A369317 <a href="/index/Ge#GF2X">Index entries for sequences operating on GF(2)[X]-polynomials</a> %F A369317 a(A129868(k)) = 2^(k+1) - 1 for any k > 0. %e A369317 The first terms, alongside the correspond GF(2)[X]-polynomials, are: %e A369317 n a(n) P(n) P(n+1) gcd(P(n), P(n+1)) %e A369317 -- ---- ----------------- ----------------- ----------------- %e A369317 1 1 1 X 1 %e A369317 2 1 X X + 1 1 %e A369317 3 1 X + 1 X^2 1 %e A369317 4 1 X^2 X^2 + 1 1 %e A369317 5 3 X^2 + 1 X^2 + X X + 1 %e A369317 6 1 X^2 + X X^2 + X + 1 1 %e A369317 7 1 X^2 + X + 1 X^3 1 %e A369317 8 1 X^3 X^3 + 1 1 %e A369317 9 3 X^3 + 1 X^3 + X X + 1 %e A369317 10 1 X^3 + X X^3 + X + 1 1 %o A369317 (PARI) a(n) = fromdigits(lift(Vec(gcd(Mod(1, 2) * Pol(binary(n)), Mod(1, 2) * Pol(binary(n+1))))), 2) %Y A369317 Cf. A091255, A129868, A369277 (distinct values), A369318 (indices of values <> 1). %K A369317 nonn,base %O A369317 1,5 %A A369317 _Rémy Sigrist_, Jan 19 2024