**21**

votes

**5**answers

937 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

**6**answers

836 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$ ...

**48**

votes

**4**answers

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 ...

**0**

votes

**1**answer

191 views

### What is a description of winning strategies in this tile game?

I'm hoping someone can help me figure out how to describe all winning strategies for "Player 1" in the following game:
Consider a board with $n$ tiles arranged in a row. Player 1 and Player 2 each ...

**6**

votes

**0**answers

401 views

### Graphs with graphic imbalance sequences

Let $G$ be simple undirected graph and $e=uv\in E(G)$.
The imbalance of the edge $e$ is the value $imb(e)=|d(u)-d(v)|$.
Let $M_{G}$ denotes the imbalance sequence (or more correctly, multiset of ...

**4**

votes

**2**answers

195 views

### Databases for sequences indexed by partitions

Is there a database for sequences indexed by partitions similar to Sloane's OEIS? I mean, I am aware that in the OEIS there are some arrays indexed by partitions, but I feel as though most of such ...

**1**

vote

**1**answer

185 views

### Nested Sequence of Integers

In some combinatorial research I came across the following nested sequence:
$$\{a_n\}=\{1,1,3,1,7,3,17,1,35,7,77,3,157,17,331,1,663,35,1361,7,2729,77,5535,3,11073, \dots\}$$
which is not in the OEIS. ...

**5**

votes

**0**answers

394 views

### Number of Configurations in the optimal Hanoi tower

There is a unique strategy how to move $n$ disks from the first rod to the second optimally and it takes $2^n-1$ steps, solution is obtained by simple recursion. I am interested into the following ...

**0**

votes

**2**answers

203 views

### sequence, such that sum of any combinations in the sequence does not equal another [closed]

Hi,
Is there any known sequence such that the sum of a combination of one subsequence never equals another subsequence sum. The subsequences should have elements only from the parent sequence.
...

**11**

votes

**0**answers

609 views

### Is OEIS A007018 really a subsequence of squarefree numbers?

A comment in A007018 a(n) = a(n-1)^2 + a(n-1), a(0)=1 claims
Subsequence of squarefree numbers (A005117). - Reinhard Zumkeller, Nov 15 2004
Is it really so?
As far as I know, it is open problem ...

**9**

votes

**2**answers

1k 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 ...

**2**

votes

**3**answers

551 views

### For any prime $p$, is there $C$ such that if $x\ge C$, then all but one integer among $x+1, x+2, \dots, x+p$ has Greatest Prime Factor $> p$

I apologize if this is a naive question about greatest prime factors (gpf). I was thinking about the sequence of integers where $\mathrm{gpf}(x) \le p$ where $p$ is any prime.
Clearly, as $x$ ...

**1**

vote

**1**answer

299 views

### Maximal difference between k randomly drawn numbers from 1 to n – Looking for formula to sequence

Hello!
I have an interesting problem that seemed simple to me, but I'm unable to solve it on my own.
Suppose I am drawing k numbers out of n numbers labeled from 1 to n.
Considering all $\binom{n}{k}$...

**3**

votes

**2**answers

926 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 $n=\mathrm {lcm} (1,2 \dots k)$, $G(n)$ ...

**-3**

votes

**2**answers

683 views

### An interesting, simple, sequence - surprised to find little material. [closed]

I've been considering this sequence:
$$1,2,3,6,12,24,48,96,192,...$$
I've generated the sequence from the rule
$$V_n=\sum_{0\leq i \lt n} V_i$$
$$V_0=1; V_1=2V_0=V_0+V_0$$
What interests me most, ...

**7**

votes

**2**answers

618 views

### Maximal number of edges and triangular cells for n points in a triangular lattice

Consider a subset of $n$ points in an equilateral triangular lattice. Draw all the edges between nearest-neighbor points.
What is the maximum, over all such subsets, of the number of edges? This ...

**21**

votes

**3**answers

1k views

### Zeroes of the random Fibonacci sequence

Let X_n be the "random Fibonacci sequence," defined as follows:
$X_0 = 0, X_1 = 1$;
$X_n = \pm X_{n-1} \pm X_{n-2}$, where the signs are chosen by independent 50/50 coinflips.
It is known that |X_n|...

**14**

votes

**2**answers

1k 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 they'...

**4**

votes

**0**answers

601 views

### Least Prime Factor in a sequence of 2n consecutive integers

I was thinking about consecutive integers and I wondered if anyone had done work exploring whether a sequence of $2n$ consecutive integers (i.e. 101,102,103,...,100+2n) always contains at least one ...

**7**

votes

**4**answers

824 views

### A Pascal's-triangle -like random process

I was exploring Pascal's triangle on a cylinder when I encountered this puzzle-like problem.
It is surely elementary, but perhaps weekend-entertaining.
Start with a permutation of $(1,2,3, \ldots, n)$...

**11**

votes

**0**answers

575 views

### Is “OEIS A001935 Number of partitions with no even part repeated” efficiently computable $\mod 4$?

Is A001935 Number of partitions with no even part repeated efficiently computable $\mod 4$?
I am interested because of this relation with sum of divisors of $8n+1$.
$\sigma(8n+1) \equiv A001935(n) \...

**8**

votes

**1**answer

715 views

### Up to $10^6$: $\sigma(8n+1) \mod 4 = OEIS A001935(n) \mod 4$ (Number of partitions with no even part repeated )

Up to $10^6$:
$\sigma(8n+1) \mod 4 = OEIS A001935(n) \mod 4$
A001935 Number of partitions with no even part repeated
Is this true in general?
It would mean relation between restricted partitions ...

**11**

votes

**1**answer

622 views

### Put as many points as possible in an equilateral triangle of side 1 with their minimal distance greater than 1/n

It is known by the pigeon-hole principle that:
If we select $5$ points within an equilateral triangle with side $1$, there must be at least two whose distance apart is less than or equal to $1/2$.
...

**5**

votes

**5**answers

618 views

### Can an integer or rational sequence satisfy some bounded order recurrence $\mod \ $ almost all primes but doesn't satisfy such in $\mathbb{Q}$?

Can an integer or rational sequence satisfy some bounded order recurrence $\mod \ $ almost all primes but doesn't satisfy such in $\mathbb{Q}$?
The recurrences $\mod p$ can be different, possibly ...

**15**

votes

**1**answer

2k views

### Conjecture on signed sum of integer fractions x/y from 1..N?

Here is a generalization of an integer challenge that was asked on Yahoo!Answers in 2009, I believe it could be original, defies induction and has exponential-complexity. Not aware of any theory that ...

**12**

votes

**11**answers

3k views

### Longest coinciding pair of integer sequences known

There are arbitrarily many pairs of integer sequences (of arbitrary origins) that coincide upto an $N$ but differ for an $n > N$. I assume, the coincidence will be considered accidentally then by ...