A317926 Denominators of rational valued sequence whose Dirichlet convolution with itself yields Euler's phi (A000010).

1, 2, 1, 8, 1, 2, 1, 16, 2, 1, 1, 8, 1, 2, 1, 128, 1, 4, 1, 4, 1, 2, 1, 16, 1, 1, 2, 8, 1, 1, 1, 256, 1, 1, 1, 16, 1, 2, 1, 8, 1, 2, 1, 8, 1, 2, 1, 128, 2, 1, 1, 4, 1, 4, 1, 16, 1, 1, 1, 4, 1, 2, 2, 1024, 1, 2, 1, 1, 1, 1, 1, 32, 1, 1, 1, 8, 1, 1, 1, 64, 8, 1, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 256, 1, 4, 2, 1, 1, 1, 1, 8, 1

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A000010(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

Mathematica

f[1] = 1; f[n_] := f[n] = (EulerPhi[n] - DivisorSum[n, f[#]*f[n/#] &, 1 < # < n &])/2; Denominator @ Array[f, 100] (* Amiram Eldar, Dec 12 2022 *)

Prog

(PARI)
A317925perA317926(n) = if(1==n, n, (eulerphi(n)-sumdiv(n, d, if((d>1)&&(d<n), A317925perA317926(d)*A317925perA317926(n/d), 0)))/2);
A317926(n) = denominator(A317925perA317926(n));

Crossrefs

Cf. A000010, A317925 (numerators).

Cf. also A046644, A317832.

Sequence in context: A317928 A011327 A318454 * A318318 A318314 A230369

Adjacent sequences: A317923 A317924 A317925 * A317927 A317928 A317929

Keyword

nonn,frac

Author

Antti Karttunen, Aug 11 2018

Status

approved