A349690 Numbers k such that the continued fraction of the abundancy index of k contains distinct elements.

1, 2, 3, 5, 6, 7, 9, 11, 12, 13, 17, 18, 19, 20, 23, 25, 27, 28, 29, 31, 33, 37, 40, 41, 43, 47, 49, 53, 56, 59, 60, 61, 67, 71, 73, 77, 79, 80, 81, 83, 88, 89, 91, 97, 101, 103, 104, 107, 109, 113, 120, 121, 125, 127, 131, 137, 139, 145, 149, 151, 155, 157, 163

Offset: 1

Author

Amiram Eldar, Nov 25 2021

Keywords

• nonn

Comments

All the primes (A000040) are terms of this sequence, since the continued fraction of the abundancy index of a prime p is {1, p}.

All the multiply-perfect numbers (A007691) are terms of this sequence, since the continued fraction of their abundancy index contains a single element.

Examples

			2 is a term since the abundancy index of 2 is 3/2 = 1 + 1/2 and the elements of the continued fraction, {1, 2}, are different.
4 is not a term since the abundancy index of 4 is 7/4 = 1 + 1/(1 + 1/3) and the elements of the continued fraction, {1, 1, 3}, are not distinct.
		

Links

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

Crossrefs

Cf. A007691, A349502, A349685, A349686.

Programs

Mathematica

c[n_] := ContinuedFraction[DivisorSigma[1, n]/n]; q[n_] := Length[(cn = c[n])] == Length[DeleteDuplicates[cn]]; Select[Range[200], q]

PARI

isok(k) = my(v=contfrac(sigma(k)/k)); #v == #Set(v); \\ Michel Marcus, Nov 25 2021