A318444 Numerators of the sequence whose Dirichlet convolution with itself yields A057660(n) = Sum_{k=1..n} n/gcd(n,k).

1, 3, 7, 35, 21, 21, 43, 239, 195, 63, 111, 245, 157, 129, 147, 6851, 273, 585, 343, 735, 301, 333, 507, 1673, 1643, 471, 3011, 1505, 813, 441, 931, 50141, 777, 819, 903, 6825, 1333, 1029, 1099, 5019, 1641, 903, 1807, 3885, 4095, 1521, 2163, 47957, 6555, 4929, 1911, 5495, 2757, 9033, 2331, 10277, 2401, 2439, 3423

Offset: 1

Author

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Keywords

• nonn
• frac
• mult

Comments

Because A057660 contains only odd values, A046644 gives the sequence of denominators. Because both of those sequences are multiplicative, this is also.

General formula: if k >= 0, m > 0, and the Dirichlet generating function is zeta(s-k)^m * f(s), where f(s) has all possible poles at points less than k+1, then Sum_{j=1..n} a(j) ~ n^(k+1) * log(n)^(m-1) * f(k+1) / ((k+1) * Gamma(m)) * (1 + (m-1)*(m*gamma - 1/(k+1) + f'(k+1)/f(k+1)) / log(n)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function. - Vaclav Kotesovec, May 10 2025

Links

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

Crossrefs

Cf. A057660, A046644 (denominators).

Cf. also A318443.

Programs

Mathematica

a57660[n_] := DivisorSigma[2, n^2]/DivisorSigma[1, n^2];
f[1] = 1; f[n_] := f[n] = 1/2 (a57660[n] - Sum[f[d]*f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
Table[f[n] // Numerator, {n, 1, 60}] (* Jean-François Alcover, Sep 13 2018 *)

PARI

up_to = 16384;
A057660(n) = sumdivmult(n, d, eulerphi(d)*d); \\ From A057660
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.
v318444aux = DirSqrt(vector(up_to, n, A057660(n)));
A318444(n) = numerator(v318444aux[n]);

PARI

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

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A057660(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A318444(k) / A046644(k) ~ n^3 * Pi^(-3/2) * sqrt(2*zeta(3)/(3*log(n))) * (1 + (1/3 - gamma/2 + 3*zeta'(2)/Pi^2 - zeta'(3)/(2*zeta(3))) / (2*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 10 2025