Integers with the same prime factors and the same aliquot sum

Number Theory Level pending

Are there two different integers $n$ and $m$ with the same prime factors (for example: $12=2^2\times 3$ and $18=2\times 3^2$ have the same prime factors 2 and 3) such that $s(n)=s(m)$ where $s(x)$ is the sum of all proper divisors of $x$ ( $s(x)=\sigma(x)-x$ , for example: $s(8)=1+2+4=7$ )?

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Moulay Abdellah El Idrissi El Mrhari
Nov 22, 2018

Let's take $n=160=2^5*5$ and $m=250=2*5^3$ , we have $s(160)=\frac{2^{5+1}-1}{2-1}*\frac{5^{1+1}-1}{2-1}-160=218=\frac{2^{1+1}-1}{2-1}*\frac{5^{3+1}-1}{2-1}-250=s(250)$ . This is the only pair bellow $10,000,000,000$ that verifies the property and I don't know if it's the only one in $\mathbb{N}$ !

Integers with the same prime factors and the same aliquot sum

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