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%I #10 Sep 13 2018 06:22:25
%S 1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,3,1,5,1,1,1,1,1,1,1,
%T 1,3,1,1,1,1,1,1,1,1,3,1,1,1,3,3,1,1,1,5,1,1,1,1,1,1,1,1,3,1,1,1,1,1,
%U 1,1,1,3,1,1,3,1,1,1,1,1,35,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,3,3,3,1,1,1,1,1
%N Numerators of the sequence whose Dirichlet convolution with itself yields A001511, the 2-adic valuation of 2n.
%H Antti Karttunen, <a href="/A318449/b318449.txt">Table of n, a(n) for n = 1..65537</a>
%F a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A001511(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%t a1511[n_] := IntegerExponent[2n, 2];
%t f[1] = 1; f[n_] := f[n] = 1/2 (a1511[n] - Sum[f[d] f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
%t Table[f[n] // Numerator, {n, 1, 105}] (* _Jean-François Alcover_, Sep 13 2018 *)
%o (PARI)
%o up_to = 65537;
%o A001511(n) = 1+valuation(n,2);
%o DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&d<n, u[d]*u[n/d], 0)))/2); u}; \\ From A317937.
%o v318449_51 = DirSqrt(vector(up_to, n, A001511(n)));
%o A318449(n) = numerator(v318449_51[n]);
%Y Cf. A001511, A318450 (denominators).
%K nonn,frac
%O 1,9
%A _Antti Karttunen_ and _Andrew Howroyd_, Aug 29 2018
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