Question

How prove this $\sum_{t|n}(d(t))^3=\left(\sum_{t|n}d(t)\right)^2$

score 445 · Answer 1 · 2026-03-25 16:02:22

445

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How prove this $\sum_{t|n}(d(t))^3=\left(\sum_{t|n}d(t)\right)^2$

Published on 25 Mar 2026 - 16:02

score 85 · Answer 2 · 2026-03-25 11:43:54

85

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When is $f(n)=\sum\limits_{d\mid n}\sigma(d)$ prime?

Published on 25 Mar 2026 - 11:43

score 296 · Answer 3 · 2014-05-24 21:23:31

296

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How to prove $D(n)<2n(\log\log n)$?

Published on 24 May 2014 - 21:23

score 44 · Answer 4 · 2014-05-30 12:39:38

44

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How to show $\sum_{n\leq x}d(n)=\sum_{ab\leq x}1$?

Published on 30 May 2014 - 12:39

score 575 · Answer 5 · 2014-05-31 11:36:54

575

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Why isn't $1$ a superior highly composite number?

Published on 31 May 2014 - 11:36

score 1.9k · Answer 6 · 2014-05-31 18:45:17

1.9k

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Why does the number of divisors of a superior highly composite number is always a highly composite number up to 720720 ? (the only exception is 120)

Published on 31 May 2014 - 18:45

score 124 · Answer 7 · 2014-06-08 10:23:26

124

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A generalization of perfect numbers

Published on 08 Jun 2014 - 10:23

score 217 · Answer 8 · 2014-06-13 18:44:14

217

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Prove that $\dfrac{\sigma_1(n)}{n} = \sigma_{-1}(n)$ where $\sigma_x(n)$ is the sum of the $x$th powers of the positive divisors of $n$.

Published on 13 Jun 2014 - 18:44

score 155 · Answer 9 · 2014-06-14 22:29:36

155

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What are the smallest numbers $n$ such that $\dfrac{d(n)}{\ln(n)} \geq k$ where $d(n) = \sigma_0(n)$ is the number-of-divisors function?

Published on 14 Jun 2014 - 22:29

score 1.1k · Answer 10 · 2014-06-20 22:25:53

1.1k

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Is there a closed form expression for the sum of all the proper divisors of an integer?

Published on 20 Jun 2014 - 22:25

score 60 · Answer 11 · 2014-06-22 14:19:34

60

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Can we prove this inequality in another way?

Published on 22 Jun 2014 - 14:19

score 173 · Answer 12 · 2014-06-22 15:12:10

173

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Prove that for any $n$ having at least two distinct prime factors, there exists a perfect square between $n$ and $\sigma(n)$

Published on 22 Jun 2014 - 15:12

score 1.4k · Answer 13 · 2014-07-18 02:00:02

1.4k

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the average order of divisor function

Published on 18 Jul 2014 - 2:00

score 489 · Answer 14 · 2014-07-20 19:32:59

489

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Infinite series involving sum of divisors function

Published on 20 Jul 2014 - 19:32

score 277 · Answer 15 · 2014-07-21 00:08:42

277

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Given an integer $n$ find smallest integer $i$ such that $σ(i)=n$. Smallest Inverse Sum of Divisors

Published on 21 Jul 2014 - 0:08

List Question

How prove this $\sum_{t|n}(d(t))^3=\left(\sum_{t|n}d(t)\right)^2$

When is $f(n)=\sum\limits_{d\mid n}\sigma(d)$ prime?

How to prove $D(n)<2n(\log\log n)$?

How to show $\sum_{n\leq x}d(n)=\sum_{ab\leq x}1$?

Why isn't $1$ a superior highly composite number?

Why does the number of divisors of a superior highly composite number is always a highly composite number up to 720720 ? (the only exception is 120)

A generalization of perfect numbers

Prove that $\dfrac{\sigma_1(n)}{n} = \sigma_{-1}(n)$ where $\sigma_x(n)$ is the sum of the $x$th powers of the positive divisors of $n$.

What are the smallest numbers $n$ such that $\dfrac{d(n)}{\ln(n)} \geq k$ where $d(n) = \sigma_0(n)$ is the number-of-divisors function?

Is there a closed form expression for the sum of all the proper divisors of an integer?

Can we prove this inequality in another way?

Prove that for any $n$ having at least two distinct prime factors, there exists a perfect square between $n$ and $\sigma(n)$

the average order of divisor function

Infinite series involving sum of divisors function

Given an integer $n$ find smallest integer $i$ such that $σ(i)=n$. Smallest Inverse Sum of Divisors

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