Question

Is $\ 7!=5040\ $ the largest highly composite factorial?

score 2.3k · Answer 1 · 2026-03-20 20:15:16

2.3k

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Is $\ 7!=5040\ $ the largest highly composite factorial?

Published on 20 Mar 2026 - 20:15

score 95 · Answer 2 · 2026-03-21 16:10:52

95

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Let $s_k$ be the $lcm$ of the numbers $1$ to $k$. Is $s_{28}$ the largest highly composite such number?

Published on 21 Mar 2026 - 16:10

score 167 · Answer 3 · 2026-03-22 08:17:55

167

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How reliable is the list of HCN-numbers considering this link?

Published on 22 Mar 2026 - 8:17

score 1k · Answer 4 · 2026-03-21 20:32:09

1k

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Prime dividing sum of divisors

Published on 21 Mar 2026 - 20:32

score 142 · Answer 5 · 2026-03-22 15:22:00

142

Views

On $-\log (2\pi)\sigma_0(n)-\frac{1}{2}\sum_{1<d\mid n}\log\left(\frac{1+2d}{(d+1)^2}\right),$ where $\sigma_0(n)=\sum_{d\mid n}1$ and its twists

Published on 22 Mar 2026 - 15:22

score 44 · Answer 6 · 2026-03-22 07:43:50

44

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Counting functions and subsets

Published on 22 Mar 2026 - 7:43

score 82 · Answer 7 · 2026-03-22 15:22:21

82

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On the sequence of positive integers satisfying $\sigma(n)\mid (n(\sigma_0(n))^2)$

Published on 22 Mar 2026 - 15:22

score 75 · Answer 8 · 2026-03-21 16:25:11

75

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The function $f(n) ={\ln(\tau(n)) \above 1.5pt \ln(2)}$ and the number $2^{\tau(n)}+1$

Published on 21 Mar 2026 - 16:25

score 187 · Answer 9 · 2026-03-22 03:39:22

187

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Prove that $\nu(n) \le \nu(2^{n}-1)$ where $\nu(n)$ is the number of positive divisors of n

Published on 22 Mar 2026 - 3:39

score 1.5k · Answer 10 · 2026-03-25 13:46:41

1.5k

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Formula for square of the number of divisors $\sum_{r\mid n} d(r^2) = d^2(n)$

Published on 25 Mar 2026 - 13:46

score 156 · Answer 11 · 2026-03-22 03:07:27

156

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Computing the bound of $d(n)$ the divisor counting "function" of $n$

Published on 22 Mar 2026 - 3:07

score 64 · Answer 12 · 2026-03-22 08:46:43

64

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Which of these two expressions (related to primes) is greater: $\frac{\sigma{(n)}}{n}$ or $\sum_{p \le n}{\frac{1}{p}}$?

Published on 22 Mar 2026 - 8:46

score 195 · Answer 13 · 2026-03-22 11:42:20

195

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Prove that: $xy = 96$ has $24$ possible integer solutions.

Published on 22 Mar 2026 - 11:42

score 326 · Answer 14 · 2026-03-22 04:42:00

326

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show that $ \sum_{n \leq z} \frac{\tau(n)}{n} = \frac{1}{2} \log^2 z + c_2 \log z + c_3 + O(\sqrt[4]{z})$

Published on 22 Mar 2026 - 4:42

score 177 · Answer 15 · 2026-03-22 11:28:41

177

Views

If $a(n)=n^2+1$ then $\gcd(a_n,2^{d(a_n)})=1\text{ or }2$?

Published on 22 Mar 2026 - 11:28

List Question

Is $\ 7!=5040\ $ the largest highly composite factorial?

Let $s_k$ be the $lcm$ of the numbers $1$ to $k$. Is $s_{28}$ the largest highly composite such number?

How reliable is the list of HCN-numbers considering this link?

Prime dividing sum of divisors

On $-\log (2\pi)\sigma_0(n)-\frac{1}{2}\sum_{1<d\mid n}\log\left(\frac{1+2d}{(d+1)^2}\right),$ where $\sigma_0(n)=\sum_{d\mid n}1$ and its twists

Counting functions and subsets

On the sequence of positive integers satisfying $\sigma(n)\mid (n(\sigma_0(n))^2)$

The function $f(n) ={\ln(\tau(n)) \above 1.5pt \ln(2)}$ and the number $2^{\tau(n)}+1$

Prove that $\nu(n) \le \nu(2^{n}-1)$ where $\nu(n)$ is the number of positive divisors of n

Formula for square of the number of divisors $\sum_{r\mid n} d(r^2) = d^2(n)$

Computing the bound of $d(n)$ the divisor counting "function" of $n$

Which of these two expressions (related to primes) is greater: $\frac{\sigma{(n)}}{n}$ or $\sum_{p \le n}{\frac{1}{p}}$?

Prove that: $xy = 96$ has $24$ possible integer solutions.

show that $ \sum_{n \leq z} \frac{\tau(n)}{n} = \frac{1}{2} \log^2 z + c_2 \log z + c_3 + O(\sqrt[4]{z})$

If $a(n)=n^2+1$ then $\gcd(a_n,2^{d(a_n)})=1\text{ or }2$?

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