cp's OEIS Frontend

A318449 Numerators of the sequence whose Dirichlet convolution with itself yields A001511, the 2-adic valuation of 2n.

%I A318449 #12 May 10 2025 04:02:46
%S A318449 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 A318449 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 A318449 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 A318449 Numerators of the sequence whose Dirichlet convolution with itself yields A001511, the 2-adic valuation of 2n.
%H A318449 Antti Karttunen, <a href="/A318449/b318449.txt">Table of n, a(n) for n = 1..65537</a>
%F A318449 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.
%F A318449 Sum_{k=1..n} A318449(k) / A318450(k) ~ n * sqrt(2/(Pi*log(n))) * (1 + (1 - gamma/2 + log(2)/2)/(2*log(n))), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, May 10 2025
%t A318449 a1511[n_] := IntegerExponent[2n, 2];
%t A318449 f[1] = 1; f[n_] := f[n] = 1/2 (a1511[n] - Sum[f[d] f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
%t A318449 Table[f[n] // Numerator, {n, 1, 105}] (* _Jean-François Alcover_, Sep 13 2018 *)
%o A318449 (PARI)
%o A318449 up_to = 65537;
%o A318449 A001511(n) = 1+valuation(n,2);
%o A318449 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 A318449 v318449_51 = DirSqrt(vector(up_to, n, A001511(n)));
%o A318449 A318449(n) = numerator(v318449_51[n]);
%Y A318449 Cf. A001511, A318450 (denominators).
%K A318449 nonn,frac
%O A318449 1,9
%A A318449 _Antti Karttunen_ and _Andrew Howroyd_, Aug 29 2018