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A318443 Numerators of the sequence whose Dirichlet convolution with itself yields A018804, Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n).

%I A318443 #18 May 10 2025 04:08:25
%S A318443 1,3,5,23,9,15,13,91,59,27,21,115,25,39,45,1451,33,177,37,207,65,63,
%T A318443 45,455,179,75,353,299,57,135,61,5797,105,99,117,1357,73,111,125,819,
%U A318443 81,195,85,483,531,135,93,7255,363,537,165,575,105,1059,189,1183,185,171,117,1035,121,183,767,46355,225,315,133,759,225,351,141,5369
%N A318443 Numerators of the sequence whose Dirichlet convolution with itself yields A018804, Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n).
%C A318443 Because A018804 gets only odd values on primes, A046644 gives the sequence of denominators. Because both of those sequences are multiplicative, this is also.
%H A318443 Antti Karttunen, <a href="/A318443/b318443.txt">Table of n, a(n) for n = 1..16384</a>
%H A318443 Vaclav Kotesovec, <a href="/A318443/a318443.jpg">Graph - the asymptotic ratio (10000 terms)</a>
%F A318443 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A018804(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%F A318443 Sum_{k=1..n} A318443(k) / A046644(k) ~ sqrt(3/2)*n^2/Pi. - _Vaclav Kotesovec_, May 10 2025
%t A318443 a18804[n_] := Sum[n EulerPhi[d]/d, {d, Divisors[n]}];
%t A318443 f[1] = 1; f[n_] := f[n] = 1/2 (a18804[n] - Sum[f[d] f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
%t A318443 a[n_] := f[n] // Numerator;
%t A318443 Array[a, 72] (* _Jean-François Alcover_, Sep 13 2018 *)
%o A318443 (PARI)
%o A318443 up_to = 16384;
%o A318443 A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
%o A318443 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 A318443 v318443aux = DirSqrt(vector(up_to, n, A018804(n)));
%o A318443 A318443(n) = numerator(v318443aux[n]);
%o A318443 (PARI) for(n=1, 100, print1(numerator(direuler(p=2, n, (1-X)^(1/2)/(1-p*X))[n]), ", ")) \\ _Vaclav Kotesovec_, May 09 2025
%Y A318443 Cf. A018804, A046644 (denominators).
%Y A318443 Cf. also A318444.
%K A318443 nonn,frac,mult
%O A318443 1,2
%A A318443 _Antti Karttunen_ and _Andrew Howroyd_, Aug 29 2018