Periodicity of $n=6$ on iterative sum power divisor function

OpenYear of origin: 1984

Posted online: 2021-07-09 11:09:08Z by Rafik Zeraoulia91

Cite as: P-210709.1

• Number Theory
• Combinatorics

Problem's Description

Let ${\sigma}_x(n) =\sum_{d\div n} d^x$ is the sum divisor function. After a few computations in wolfram alpha about the divisor function for some values of $n$ to look the behavior of $\sigma_x(n)\bmod n$ for $\,n=6,\,$ we got this result : $\sigma_x(6)=0 \bmod 6$ for $x$ odd and $2 \bmod 6$ if $x$ is even. One can ask this question: Is $n=6$ the only integer satisfies $\sigma_x(n) \equiv 0\bmod n$ for every odd integer $x > 0$ and $2 \bmod n$ if $x$ is even integer? One can answer this question using divisibility and Congruence properties for sum power divisor function, you can see the proof in this paper. The answer of this question yield to reformulate the following problem regarding periodicity on iterative of sum power divsior function.

Problem: Is $n=6$ the only integer satisfy periodicity on iterative of sum power divisor function with small prime period namely $L=2$?

The challenging problem is that the analysis had some flaws. However, one expects multiperfect numbers other than $1$ and $6$ to be a multiple of $4$; when $n$ satisfies $\sigma(n) \bmod n = 0$ and $\sigma_2(n) \bmod n = 2$, and in addition $n \bmod 4 = 0$, then all odd prime factors of $n$ except one must occur to an even multiplicity, and the remaining odd prime factor must occur to a multiplicity of $1 \bmod 4$ and must be a prime that is $3 \bmod 4$. While simple, these observations say a lot about $n$ and suggest that any numbers satisfying the title congruences are rare indeed, perhaps more so than odd multiperfect numbers, For more informations about the attempt which is given by Gerhard Paseman you may check this MO answer.

Note: The motivation behind solving the problem is to add some properties for aliquot sequences and to find a new equivalence to the Riemann Hypothesis.

arXiv

2. ArticleIs an originIterating the Sum-of-Divisors Function

Experimental Mathematics Volume 5 (No 2), 1-10, 1996

3. ArticleIs an originOn the third iterates of the $\phi$ and $\sigma$ functions

Colloquium Mathematicum XLIX (0010-1354), 123-130, 1984