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First column of A115872.
6

%I #7 Jan 04 2022 18:35:24

%S 1,1,3,1,7,3,7,1,15,7,3,3,5,7,15,1,31,15,7,7,31,3,7,3,31,5,31,7,31,15,

%T 31,1,63,31,15,15,7,7,7,7,63,31,3,3,63,7,15,3,21,31,63,5,63,31,7,7,9,

%U 31,63,15,21,31,63,1,127,63,31,31,15,15,15,15,127,7,31,7,15,7,15,7,51

%N First column of A115872.

%F a(2^k) = 1, a(2n) = a(n).

%t X[a_, b_] := Module[{A, B, C, x},

%t A = Reverse@IntegerDigits[a, 2];

%t B = Reverse@IntegerDigits[b, 2];

%t C = Expand[

%t Sum[A[[i]]*x^(i - 1), {i, 1, Length[A]}]*

%t Sum[B[[i]]*x^(i - 1), {i, 1, Length[B]}]];

%t PolynomialMod[C, 2] /. x -> 2];

%t T[n_, k_] := Module[{x = BitXor[n - 1, 2 n - 1], k0 = k},

%t For[i = 1, True, i++, If[n*i == X[x, i],

%t If[k0 == 1, Return[i], k0--]]]];

%t a[n_] := T[n, 1];

%t Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Jan 04 2022 *)

%Y Cf. A114395, A114396, A115872.

%K nonn

%O 1,3

%A _Antti Karttunen_, Feb 07 2006