A349502 -id:A349502 - cp's OEIS frontend

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

Amiram Eldar, Nov 25 2021

nonn

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.

			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.

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

Cf. A007691, A349502, A349685, A349686.

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

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

A349503 a(n) is the least number k such that the continued fraction of the harmonic mean of the divisors of k contains n elements that are all distinct.

Original entry on oeis.org

1, 2, 20, 52, 156, 768, 8244, 25808, 406764, 3610688, 41395016, 453695175, 3325792768

Offset: 1

Amiram Eldar, Nov 20 2021

			The elements of the continued fractions of the harmonic mean of the divisors of the first 13 terms:
   n        a(n)  elements
  --  ----------  -----------------------------
   1           1  1
   2           2  1,3
   3          20  2,1,6
   4          52  3,5,2,4
   5         156  4,1,3,2,5
   6         768  6,1,3,4,2,13
   7        8244  7,11,8,3,1,13,2
   8       25808  5,6,3,13,1,2,4,7
   9      406764  7,8,3,6,9,2,1,4,12
  10     3610688  7,18,5,2,3,6,1,4,13,11
  11    41395016  7,1,12,8,4,2,3,5,19,6,10
  12   453695175  16,5,8,1,10,48,7,13,2,3,6,4
  13  3325792768  19,1,21,7,6,3,12,13,5,9,2,8,4

Cf. A349473, A349502.

cflen[n_] := Module[{cf = ContinuedFraction[DivisorSigma[0, n]/DivisorSigma[-1, n]], len}, If[(len = Length[cf]) == Length[DeleteDuplicates[cf]], len, 0]]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = cflen[n]; If[i > 0 && i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; TakeWhile[s, # > 0 &]]; seq[10, 10^7]

cp's OEIS Frontend

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

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