**14**

votes

**1**answer

901 views

### Some unpublished notes of Hofstadter

I'm looking for some unpublished notes called "Eta Lore," which are apparently related to a talk Douglas Hofstadter first gave at the Stanford Math Club in 1963. I know these notes exist because ...

**39**

votes

**4**answers

2k views

### Strange (or stupid) arithmetic derivation

Let us consider the following operation on positive integers: $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \qquad f(n):= \prod_{i=1}^{k}\alpha_ip_i^{\alpha_i-1}$$ (Is it true that if we apply this operation to ...

**20**

votes

**5**answers

635 views

### Sequences with integral means

Let $S(n)$ be the sequence whose first element is $n$, and from then onward,
the next element is the smallest natural number ${\ge}1$ that ensures that the
mean of all the numbers in the sequence is ...

**3**

votes

**3**answers

749 views

### Which $n$ maximize $G(n)=\frac{\sigma(n)}{n \log \log n}$?

By Robin's theorem
$$G(n)=\frac{\sigma(n)}{n \log \log n}$$
is bounded by $e^\gamma \approx 1.78107241799$ for $n>5040$ assuming Riemann hypothesis .
For $n=\mathrm {lcm} (1,2 \dots k)$, $G(n)$ ...

**13**

votes

**6**answers

670 views

### Is the sequence $a_n=c a_{n-1} - a_{n-2}$ always composite for $n > 5$?

Numerical evidence suggests the following.
For $c \in \mathbb{N}, c > 2$ define the sequence $a_n$ by
$a_0=0,a_1=1, \; a_n=c a_{n-1} - a_{n-2}$
For $ 5 < n < 500, \; 2 < c < 100$ ...

**13**

votes

**1**answer

421 views

### Are the asymptotics of A003238 known?

Sequence A003238 of the OEIS counts ``rooted trees with $n$ vertices in which vertices at the same level have the same degree.'' The sequence, $a$, begins
1, 1, 2, 3, 5, 6, 10, 11, 16, ...
and it is ...