[seqfan] Re: Integers with a peculiar divisibility property

[Allan Wechsler]

I must have had a bug in my original code, since everybody agrees that my
results were consistent with divisibility of sigma(sigma(n^4)) by n, and
not by n^2.

I shouldn't be sequencing during work hours, so I will get to this this
evening perhaps. But if Sean or Harvey or Georg or Konstantinos want to
jump in and create one or both sequences, I will not feel at all slighted!

There are many dimensions of possible generalization available. In general
we want to know for which n sigma^K (n^L) is divisible by n^M. For each
choice of K, L, and M, we might be specially interested in the primitive
elements (not products of smaller elements), or in the prime elements
(which is why I noticed the intriguing case of sigma^2(19^4) in the first
place).

In case anybody is curious, this question comes up when hunting multiply
perfect numbers. The 19^4 case gives rise to Poulet's Substitution, which
enabled Paul Poulet to discover several dozen multiply perfect numbers in
1925 (according to Wikipedia) or in 1929 (according to Achim Flammenkamp).

On Mon, Jan 7, 2019 at 6:37 AM Konstantinos Vlachopoulos via SeqFan <
seqfan at list.seqfan.eu> wrote:

> Here is some fairly fast Python code. It uses sympy (an external library):
>
> from sympy import divisors
>
>
> def sigma(n):
>     return sum(divisors(n))
>
>
> def calc(n):
>     return sigma(sigma(n**4)) % n**2
>
>
> def test(upper, lower=1):
>     for n in range(lower, upper):
>         if calc(n) == 0:
>             print(n)
>
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