[seqfan] Some progress in k-imperfect numbers (A127724)

Allan Wechsler acwacw at gmail.com
Thu Feb 13 00:24:47 CET 2020

```The sequence  https://oeis.org/A127724 collects numbers n such that
k*rho(n) = n, where rho is an arithmetic function that can be described as
an alternating sum of the divisors of n.
Example: rho(12) = 12 - 6 - 4 + 2 + 3 - 1 = 6, and 12 = 6*2, so 12 is
2-imperfect.

Yesterday afternoon I discovered the first known 5-imperfect number, the
208-digit number shown below.

1947793410288108579327587698415272737289992039373107522449638016140636142596276017072442826236838486130285072853813458640948347868163450516845165654669897619318450994951647517899051394662400000000000000000000

Up until then, we did not know if there were any 5-imperfect numbers.
Michel Marcus was kind enough to verify the correctness of this example,
and I will be adding it to the commentary on A127724. (I cannot add it to
the sequence, because we are fairly sure we don't know all the k-imperfect
numbers smaller than it.) Additional verification would of course be
welcome.

I am contemplating writing a small paper sketching the search technique
that yielded this result.

```