A342866 The number of elements in the continued fraction for phi(n)/n, where phi is the Euler totient function (A000010).

1, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 4, 3, 3, 2, 3, 3, 3, 3, 3, 4, 3, 2, 6, 3, 5, 2, 3, 3, 6, 3, 3, 3, 3, 3, 4, 3, 3, 2, 3, 3, 6, 3, 3, 2, 5, 3, 6, 3, 3, 4, 3, 3, 4, 2, 7, 4, 3, 3, 6, 4, 3, 2, 3, 3, 4, 3, 6, 3, 3, 3, 3, 3, 3, 3, 4, 3, 6

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = 2 if and only if n is in A007694.

a(p) = 3 for an odd prime p.

Example

a(2) = 2 since the continued fraction of phi(2)/2 = 1/2 = 0 + 1/2 has 2 elements: {0, 2}.
a(3) = 3 since the continued fraction of phi(3)/3 = 2/3 = 0 + 1/(1 + 1/2) has 3 elements: {0, 1, 2}.
a(15) = 4 since the continued fraction of phi(15)/15 = 8/15 = 0 + 1/(1 + 1/(1 + 1/7)) has 4 elements: {0, 1, 1, 7}.

Mathematica

a[n_] := Length @ ContinuedFraction[EulerPhi[n]/n]; Array[a, 100]

Prog

(PARI) a(n) = #contfrac(eulerphi(n)/n); \\ Michel Marcus, Mar 30 2021

Crossrefs

Cf. A000010, A007694, A076512, A109395, A342867.

Cf. A071862 (similar, with sigma(n)/n).

Sequence in context: A139713 A171465 A178620 * A023524 A068639 A074070

Adjacent sequences: A342863 A342864 A342865 * A342867 A342868 A342869

Keyword

nonn,easy

Author

Amiram Eldar, Mar 27 2021

Status

approved