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

1, 3, 5, 23, 9, 15, 13, 91, 59, 27, 21, 115, 25, 39, 45, 1451, 33, 177, 37, 207, 65, 63, 45, 455, 179, 75, 353, 299, 57, 135, 61, 5797, 105, 99, 117, 1357, 73, 111, 125, 819, 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

Offset: 1

Author

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Keywords

• nonn
• frac
• mult

Comments

Because A018804 gets only odd values on primes, A046644 gives the sequence of denominators. Because both of those sequences are multiplicative, this is also.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Crossrefs

Cf. A018804, A046644 (denominators).

Cf. also A318444.

Programs

Mathematica

a18804[n_] := Sum[n EulerPhi[d]/d, {d, Divisors[n]}];
f[1] = 1; f[n_] := f[n] = 1/2 (a18804[n] - Sum[f[d] f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
a[n_] := f[n] // Numerator;
Array[a, 72] (* Jean-François Alcover, Sep 13 2018 *)

PARI

up_to = 16384;
A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
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&&dA317937.
v318443aux = DirSqrt(vector(up_to, n, A018804(n)));
A318443(n) = numerator(v318443aux[n]);

PARI

for(n=1, 100, print1(numerator(direuler(p=2, n, (1-X)^(1/2)/(1-p*X))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A018804(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A318443(k) / A046644(k) ~ sqrt(3/2)*n^2/Pi. - Vaclav Kotesovec, May 10 2025