# Questions tagged [divisor-sum]

For questions on the divisor sum function and its generalizations.

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### How did Ramanujan came up with this?

The following is a picture of equation from Ramanujan's lost notebook. In this page, Ramanujan gives a closed form for, $$\sum_{n\geq 1}\sigma_{s}(n)x^{n}$$ In an attempt initially it's claimed that, ...
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### Does the following lower bound improve on $I(q^k)+I(n^2) > \frac{57}{20}$, where $q^k n^2$ is an odd perfect number?

Preamble: This question is an offshoot of this earlier post. (This inquiry has likewise been cross-posted to MO last June $10, 2022$.) Let $N = q^k n^2$ be an odd perfect number with special prime $q$...
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### Finding the Ceiling function of $(\sqrt{3}+ 1)^{2n}$

Ceiling function of $(\sqrt{3}+ 1)^{2n}$ is $(\sqrt{3} + 1)^{2n} + (\sqrt{3} - 1)^{2n}$. While solving a problem that states $2^{n+1}$ divides ceiling function of $(\sqrt{3}+ 1)^{2n}$. I went through ...
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### Does $I = \gcd(n,\sigma(n^2)) = (\frac{n}{\sigma(q^k)/2})\cdot\gcd(\sigma(q^k)/2,n)$ imply that $\sigma(q^k)/2 \mid n$ holds?
Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Define the GCDs: $$G = \gcd\bigg(\sigma(q^k),\sigma(n^2)\bigg)$$ H = \...