Revision 54e5c851-e550-4d40-b31f-00922b3e09aa - Code Golf Stack Exchange

# [Charcoal], 53 bytes

 Ｎθ≔⁰ζＦＮ«≔⟦⟧υＷ⁻θ⊕⌕υζ«≦⊕ζ≔ζη≔⟦⟧υＦθ«≔↨Φη∧μ¬﹪ημ¹η⊞υη»»»Ｉζ

[Try it online!][TIO-km5bd9ru] Link is to verbose version of code. Takes `n` and `m` as inputs and outputs the `m`th number, including duplicate sequences. Very slow so in practice you need `n<3`. Explanation:

 Ｎθ

Input `n`.

 ≔⁰ζ

Start at zero.

 ＦＮ«

Repeat `m` times.

 ≔⟦⟧υ

Clear any previous calculation.

 Ｗ⁻θ⊕⌕υζ«

Repeat until a perfect `n`-cycle is found.

 ≦⊕ζ

Increment the test number.

 ≔ζη

Make a copy of the number.

 ≔⟦⟧υ

Start building a list of divisor sums.

 Ｆθ«

Repeat `n` times.

 ≔↨Φη∧μ¬﹪ημ¹η

Calculate the next divisor sum.

 ⊞υη

Save it to the list.

 »»»Ｉζ

Output the final result.

[Charcoal]: https://github.com/somebody1234/Charcoal
[TIO-km5bd9ru]: https://tio.run/##TY69CsIwGEXn9Cm@8QtE8Gd0UkFwqLiLQ2yjCaSJzY@C4rPH1FopZLnJzTm3ktxVluuUduYWwz42Z@Gwpcti5b26GpwyeOZ0sQ5wXKEUXgX5lY4nBjG3yEMqLQBLZaLHlsHOVE40wgRR41aZGmOHo/1nUvLbDzDq9T4yoJ8M5Dj/VeQ7qe1Jw@uae5FFOuSFksEqGxsGexuwtHXUtrtsOj@DGR3I5BC97Jb18V3k8y4OTpmAG@4D5sXLlOawSJO7/gA "Charcoal – Try It Online"