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

1,2

Comments

Because A057660 contains only odd values, 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

Formula

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

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 *)

Prog

(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&&d<n, u[d]*u[n/d], 0)))/2); u}; \\ From A317937.
v318444aux = DirSqrt(vector(up_to, n, A057660(n)));
A318444(n) = numerator(v318444aux[n]);

Crossrefs

Cf. A057660, A046644 (denominators).

Cf. also A318443.

Sequence in context: A192880 A355156 A365140 * A334314 A179115 A299300

Adjacent sequences: A318441 A318442 A318443 * A318445 A318446 A318447

Keyword

nonn,frac,mult

Author

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Status

approved