For questions about sequences of integers. References are often made to the online resource oeis.org.

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14
votes
1answer
949 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Some unpublished notes of Hofstadter

I'm looking for some unpublished notes called "Eta Lore," which are apparently related to a talk Douglas Hofstadter first gave at the Stanford Math Club in 1963.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 ...
49
votes
4answers
4k 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 ...
8
votes
2answers
870 views

sequences with a fractal dimension

This is inspired by the self-similarity of the celebrated Golay-Rudin-Shapiro sequence, more exactly, of its alternating partial sums. (This latter one is oeis 020990). The pictures show the 550 first ...
20
votes
5answers
639 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 ...
13
votes
6answers
695 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$ ...
3
votes
3answers
787 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
1answer
451 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 ...