A349474 a(n) is the length of the continued fraction of the harmonic mean of the divisors of n.

1, 2, 2, 4, 3, 1, 3, 3, 2, 3, 3, 4, 3, 2, 2, 7, 3, 4, 3, 3, 5, 3, 3, 2, 6, 3, 4, 1, 3, 2, 3, 2, 3, 4, 3, 8, 3, 4, 5, 4, 3, 2, 3, 2, 3, 4, 3, 5, 6, 4, 3, 4, 3, 4, 2, 5, 5, 7, 3, 3, 3, 5, 7, 7, 3, 3, 3, 3, 3, 3, 3, 4, 3, 5, 7, 4, 4, 4, 3, 4, 6, 6, 3, 2, 4, 6, 3

Offset

1,2

Comments

a(n) = 1 if and only if n is a harmonic number (A001599).

a(n) <= 2 if and only if n is in A348865.

Links

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

Example

a(1) = 1 since the harmonic mean of the divisors of 1 is 1 and its continued fraction has 1 element, {1}.
a(2) = 2 since the harmonic mean of the divisors of 2 is 4/3 = 1 + 1/3 and its continued fraction has 2 elements, {1, 3}.
a(4) = 4 since the harmonic mean of the divisors of 4 is 12/7 = 1 + 1/(1 + 1/(2 + 1/2)) and its continued fraction has 4 elements, {1, 1, 2, 2}.

Mathematica

a[n_] := Length @ ContinuedFraction[DivisorSigma[0, n] / DivisorSigma[-1, n]]; Array[a, 100]

Crossrefs

Row length of A349473.

Cf. A001599, A071862, A099377, A099378, A348865.

Sequence in context: A252040 A084896 A011388 * A105114 A166284 A098086

Adjacent sequences: A349471 A349472 A349473 * A349475 A349476 A349477

Keyword

nonn

Author

Amiram Eldar, Nov 19 2021

Status

approved