# [seqfan] Re: oddness of iterations of sigma()

Thu Nov 14 12:26:34 CET 2013

```I observated a possible identity: if p is odd (not necessarily is prime), then
p+p^2+p^3+p^4+((p-1)/2)^2=
(ceil(sqrt(1+p+p^2+p^3+p^4)))^2.
I say "possible", since yet I have not proved it, but
by handy I verified it for many p, including , e.g., p=203.
If , indeed, it is identity, than, at least, for even s the pairs (r=4, arbitrary primes p>3) are impossible.

Regards,

----- Original Message -----
From: Donovan Johnson <donovan.johnson at yahoo.com>
Date: Wednesday, November 13, 2013 17:20
Subject: [seqfan]  oddness of iterations of sigma()
To: Seqfan <seqfan at list.seqfan.eu>

> 3. Note that sigma(3^4) = 11^2. Does there exists another pair (p,r)
> such that p is prime, r > 1 and sigma(p^r) = q^s where q is
> prime and
> s > 1?
>
>
> Regarding question #3:
> sigma(3^4) = 11^2 is the only solution for p^r < 10^16.
>
> Donovan
>
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