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A349690
Numbers k such that the continued fraction of the abundancy index of k contains distinct elements.
2
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
1,2
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.
LINKS
EXAMPLE
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.
MATHEMATICA
c[n_] := ContinuedFraction[DivisorSigma[1, n]/n]; q[n_] := Length[(cn = c[n])] == Length[DeleteDuplicates[cn]]; Select[Range[200], q]
PROG
(PARI) isok(k) = my(v=contfrac(sigma(k)/k)); #v == #Set(v); \\ Michel Marcus, Nov 25 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Nov 25 2021
STATUS
approved