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A318450 Denominators of the sequence whose Dirichlet convolution with itself yields A001511, the 2-adic valuation of 2n.

%I A318450 #12 Sep 13 2018 06:22:16
%S A318450 1,1,2,1,2,2,2,1,8,2,2,2,2,2,4,1,2,8,2,2,4,2,2,2,8,2,16,2,2,4,2,1,4,2,
%T A318450 4,8,2,2,4,2,2,4,2,2,16,2,2,2,8,8,4,2,2,16,4,2,4,2,2,4,2,2,16,1,4,4,2,
%U A318450 2,4,4,2,8,2,2,16,2,4,4,2,2,128,2,2,4,4,2,4,2,2,16,4,2,4,2,4,2,2,8,16,8,2,4,2,2,8
%N A318450 Denominators of the sequence whose Dirichlet convolution with itself yields A001511, the 2-adic valuation of 2n.
%C A318450 The sequence seems to give the denominators of several other similarly constructed "Dirichlet Square Roots".
%H A318450 Antti Karttunen, <a href="/A318450/b318450.txt">Table of n, a(n) for n = 1..16384</a>
%F A318450 a(n) = denominator 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.
%F A318450 a(n) = 2^A318451(n).
%t A318450 a1511[n_] := IntegerExponent[2n, 2];
%t A318450 f[1] = 1; f[n_] := f[n] = 1/2 (a1511[n] - Sum[f[d] f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
%t A318450 Table[f[n] // Denominator, {n, 1, 105}] (* _Jean-François Alcover_, Sep 13 2018 *)
%o A318450 (PARI)
%o A318450 up_to = 65537;
%o A318450 A001511(n) = 1+valuation(n,2);
%o A318450 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 A318450 v318449_51 = DirSqrt(vector(up_to, n, A001511(n)));
%o A318450 A318450(n) = denominator(v318449_51[n]);
%Y A318450 Cf. A001511, A318449 (numerators), A318451.
%K A318450 nonn,frac
%O A318450 1,3
%A A318450 _Antti Karttunen_ and _Andrew Howroyd_, Aug 29 2018