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

1,2

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)

Formula

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.

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

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 *)

Prog

(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&&d<n, u[d]*u[n/d], 0)))/2); u}; \\ From A317937.
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

Crossrefs

Cf. A018804, A046644 (denominators).

Cf. also A318444.

Sequence in context: A064187 A112686 A380905 * A272426 A270180 A270793

Adjacent sequences: A318440 A318441 A318442 * A318444 A318445 A318446

Keyword

nonn,frac,mult

Author

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Status

approved