**2**

votes

**1**answer

103 views

### GCD for two Cullen numbers

The $n$'th Cullen number is $C_n = n\cdot2^n+1$.
If $m$ and $n$ are natural numbers, what can one say about $\gcd(C_n,C_m)$, where $m$ and $n$ are different positive integers?

**-2**

votes

**0**answers

23 views

### weighted restricted integer compositions and extended binomial coefficients [on hold]

proof of
d_{S,f}(n,k) = \binom{k}{n}{(f(s)){s\in S}}

**4**

votes

**1**answer

103 views

### Betweenness in permutations

Let us consider permutations $\pi$ on $\{1,\dots,n\}$ as sequences $\pi(1),\pi(2),\dots,\pi(n)$. For a permutation $\pi$ let $R(\pi)$ be the ternary relation with $(x,y,z)\in R(\pi)$ whenever element $...

**0**

votes

**0**answers

375 views

### Number Theory and d-Self-Contained Numbers

Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$, let us associate to it the set $S_{N} = \bigcup_{j=1}^{n}\{(a_{j},j)\}$. We're going to define a d-self-contained number as any natural number ...

**0**

votes

**0**answers

184 views

### Number Theory and p-Power-Partitioned Numbers

Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$, we're going to define its digits-partition as the next set $D_{N} = \bigcup_{j=1}^{n}\bigcup_{k=1}^{p(a_{j})}\{(P_{k},j)\}$, where each pair $(...

**7**

votes

**2**answers

458 views

### The p-adic valuation of a linear recurrence

Let $(u_n)_{n \geq 0}$ be an integer-valued linear recurrence of order $k \geq 1$. Precisely,
$$u_n = a_1 u_{n-1} + \cdots + a_k u_{n - k} \quad \forall n \geq k ,$$
for some $a_1, \ldots, a_k \in \...

**1**

vote

**0**answers

46 views

### The connection between the length of Fibonacci $p$-step numbers and it's limit values

One of the most important generalization of the classical Fibonacci numbers is the Fibonacci $p$-step numbers that is defined as follows
\begin{equation}\label{cp26}
F_n^{(p)}=F_{n-1}^{(p)}+F_{n-2}^...

**2**

votes

**1**answer

245 views

### Number of subsets that sum to $0$

Suppose you choose $n$ distinct random numbers from a contiguous subset of cardinality $f({\beta, n})$ with at least $f({\alpha_+, n})$ positive and at least $f({\alpha_-, n})$ negative values from a ...

**1**

vote

**1**answer

189 views

### Repdigit numbers, which are sum of consecutive squares

Following up on this question,
http://math.stackexchange.com/questions/1788015/is-112122132142152162-1111-special/1788102?noredirect=1#comment3649733_1788102
is anything known about the sequence of ...

**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'...

**8**

votes

**1**answer

241 views

### Arbitrarily many primes in a Fibonacci-type sequence

It is conjectured that the standard Fibonacci sequence contains infinitely many primes. While this is perhaps too difficult, I am wondering about the following simpler version:
Question. For any $K$, ...

**38**

votes

**2**answers

1k views

### Can we find lattice polyhedra with faces of area 1,2,3,…?

I asked this question two months ago on MSE, where it earned the rare
Tumbleweed badge for garnering zero votes, zero answers, and 25 views over 61 days.
Perhaps justifiably so! Here I repeat it with ...

**7**

votes

**1**answer

89 views

### Integer Recursion Reference Request

I've run across the following recursion which at times seems very steady and predictable and at other times seems very chaotic.
Let $c_1, \dots c_k, b_0, m \in \mathbb{Z}$ with $b_0>m\ge 3$ and $...

**26**

votes

**1**answer

538 views

### Why do the adjoint representations of three exceptional groups have the same first eight moments?

For a representation of a compact Lie group, the $n$th moment of the trace of that representation against the Haar measure is the dimension of the invariant subspace of the $n$th tensor power. The ...

**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 ...

**8**

votes

**1**answer

716 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 ...

**9**

votes

**2**answers

766 views

### Consecutive numbers with mutually distinct exponents in their canonical prime factorization

Is it possible to find 23 consecutive positive integers each of which has mutually distinct exponents in its canonical prime factorization? Such numbers are sequence A130091 in OEIS. 24 such numbers ...

**0**

votes

**1**answer

223 views

### 4th Order Floretions: Floret's Equation [closed]

Update: I've marked this question as answered. If you are thinking "What the heck are floretions?", go right to the answer provided by the Grinch. I definitely should have added clearer information on ...

**7**

votes

**3**answers

276 views

### Upper bound on length of addition chain

An addition chain for $n$ is a finite sequence of integers starting at 1 and ending at $n$, such that each element is a sum of two previous elements. A short addition chain for $n$ can be used, for ...

**1**

vote

**0**answers

115 views

### Is there a “complete” Sidon sequence?

A sequence of natural numbers $(a_n)$ with the property that all pairwise sums of elements are distinct is called a Sidon sequence and it is proved there are at most $s(n)\sim\sqrt n$ elements of ...

**2**

votes

**3**answers

215 views

### Making integer multisets graphic

Let $M=(X,f)$ be a multiset, where $X$ is the underlying set of elements and $f:X\rightarrow\mathbb{N}$ is the multiplicity function. For every $k\in\mathbb{N}$ put $k\cdot M:=(X,k\cdot f)$. It is ...

**4**

votes

**0**answers

168 views

### A strange polynomial equality

In my answer to this question, I have obtained that the polynomial $p(x)$ of degree $2n$ with nonnegative values on $[-1,1]$ with $p(\pm1)=1$ has $\int_{-1}^1 p(x)\,dx\geq \frac{4}{(n+1)(n+2)}$, and ...

**4**

votes

**4**answers

522 views

### What is the motivation and purpose of the Floretion group?

When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their ...

**2**

votes

**0**answers

115 views

### Additive combinatorics and a Diophantine equation

Let $(n_k)_{1 \leq k \leq N}$ be a sequence of distinct positive integers. For $v \in \mathbb{Z}$ set
$$
A_N(v) = \# \Big\{ (k,\ell) \in \{1, \dots, N\}^2, ~k \neq \ell:\quad n_k - n_\ell = v \Big\}.
$...

**5**

votes

**1**answer

496 views

### Intuition behind salient numbers in number of h-cobordism classes of smooth homotopy n-spheres

The Wikipedia article on Exotic Sphere displays this sequence of numbers (see also OEIS A001676 and the Milnor link therein) for the order of the classses as
$$1, \;1, \;1,\; 1,\; 1, \;1, \;28,\; 2,\;...

**3**

votes

**0**answers

58 views

### Cardinal of a set cinsist of product of two sets?

Let
$$
A=\{1,2,\ldots,p-1\},\qquad B=\{1,2,\ldots,q-1\}
$$
where $p,q$ are primes not necessarily distinct.
Is there any elementary way to find the cardinal of the following set
$$
AB=\{ab:\ a\in A,\ ...

**1**

vote

**2**answers

136 views

### Values of the Kronecker symbol of recursvie sequences

Is there something known about the values of the Kronecker symbol $\left(\frac{a_n}{a_{n+1}}\right)$ if $a_n$ is a recursive sequence? I know something about this if $a_n$ is the Fibonacci sequence or ...

**11**

votes

**1**answer

241 views

### A Collatz-like question about permutations

An answer to this question would provide an explicit counterexample to this question, but otherwise I don't know if it is interesting.
Consider all permutations $\pi$ on the natural numbers such that ...

**0**

votes

**1**answer

95 views

### How to write a given rank matrix with some constraints?

If I want to write an $m\times n$ $0/1$ matrix with only rows or columns distinct, I could just pick $m$ or $n$ distinct natural numbers effectively writing them down as rows or columns in base $2$.
...

**6**

votes

**1**answer

297 views

### Conway's subprime Fibonacci sequences

I want to be certain I have the latest information on
Conway's subprime Fibonacci sequences,
arXiv-posted a year ago; I am referencing the status in
a review.
To wit, starting with $(0,1)$:1
$$
0, 1, ...

**3**

votes

**0**answers

51 views

### Increasing integral sequence of intermediate growth which is periodic modulo almost all primes

Many integral sequences are periodic modulo (almost) all primes.
However all examples I know are either evaluations of suitable polynomials on consecutive integers (trivial examples) or grow at least ...

**3**

votes

**0**answers

84 views

### Searching information on a certain function with a fixed point property connecting Moebius $\mu$ and Fibonacci numbers

Let $\mu$ be the Moebius function and define for $1\leq n\in\mathbb{N}$
$$
f(n) =
\left\{
\begin{array}{ll}
\mu\left(\frac{n}{2}\right) + \mu\left(\frac{n}{4}\right), & n\equiv 0, 4, 8\mod 12, \\
...

**0**

votes

**0**answers

95 views

### Classic question on integer partitions (with distinct summands)

I guess that the following was solved sometime in the 18th century, but could not find a reference to it. I am interested in approximations to the following integer partition problem:
Denote $R(N,L)$ ...

**0**

votes

**1**answer

175 views

### Number of squares in a grid under certain conditions

Consider an $(n+1)\times (n+1)$ grid of lattice points in the plane.
$A(n):$ # of squares with vertices on the grid.
It's relatively well-known that $A(n)=\frac{n(n+1)^2(n+2)}{12}$. Now, $A(n) = B(...

**6**

votes

**1**answer

165 views

### On one class of Somos-like sequences

This question is motivated by integrability of the sequence mistakenly arisen in the question Does this sequence always give an integer?
Let $m_1,\ldots, m_{k-1}$ be positive integers and sequence $\{...

**2**

votes

**0**answers

155 views

### Avoiding Fibonacci-like sequences

Suppose we are trying to avoid 3-term arithmetic progressions. There are two relevant sequences in the OEIS pertaining to this:
A003278: The sequence whose $n^{\text{th}}$ term is the smallest number ...

**12**

votes

**0**answers

342 views

### Word complexity of primes mod 4

For an infinite binary word $w$, the word complexity $f_w(n)$ is defined as the number of different subwords of length $n$. The asymptotic behavior of this function is an important parameter of the ...

**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 ...

**3**

votes

**0**answers

231 views

### What are the values of this sequence?

Let $F_n$ denote the $n$th Fibonacci number.
Then $\prod\limits_{i=1}^{\infty}(1-x^{F_i})$ is a series all of whose coefficients are either $-1$, $0$ or $+1$.
The sequence of the coefficients in ...

**36**

votes

**1**answer

961 views

### Mod sequences that seem to become constant; and the number 316

Define a "mod sequence" of nonnegative integers
based on one start parameter $s$, its first term,
as follows.
$A(s)=(a_1,a_2,\ldots,a_n,\ldots)$
with $a_1 = s$
and
$$ a_n = \left(\sum_{k=1}^{n-1} a_k \...

**4**

votes

**1**answer

182 views

### Is $p$ is square modulo $F_p$ when $p=4k+1 > 5$?

$F_n$ are the Fibonacci numbers.
In On computing factors of cyclotomic polynomials p.1 for odd square-free $n>1$ the cyclotomic polynomial $\Phi_n(x)$
satisfies:
$$ 4 \Phi_n(x)=A_n(x)^2 - (-1)^{(...

**2**

votes

**1**answer

157 views

### Tower-of-squares sequence divides linear recurrent A001921 sequence?

Let $(a_n)$ be the A001921 sequence
$$
a_0 = 0,\ a_1 = 7, \quad a_{n+2} = 14a_{n+1} - a_n + 6.
$$
Let $(b_k)$ be the (almost)"tower-of-squares" sequence defined by
$$
b_0=2, \quad b_{k+1}=2b_k^...

**6**

votes

**0**answers

138 views

### When is the ratio of Jacobi theta functions algebraic?

Probably this is well known.
$\theta_2$ and $\theta_3$ are Jacobi theta functions
as defined in mathworld (31) and (32).
For natural $n$ define
$$ f(n) = \frac{\theta_2(-e^{-\pi\sqrt{n}})}{\theta_3(-...

**7**

votes

**1**answer

412 views

### More asymptotics for trees

This is a follow up to my recent question on the asymptotics of A003238. Lucia gave a fine answer to that question, but as I hinted the 'real' problem I have in mind is slightly different, and I've ...

**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 ...

**5**

votes

**1**answer

524 views

### Arbitrarily large $n$ divides $F_n$

Is it true that there exists $n \in \mathbb{N}$ with arbitrarily many prime factors such that $n$ divides $F_n$, where $F_n$ represents the n-th Fibonacci number?

**5**

votes

**1**answer

360 views

### Flow of an integer

I've stumbled across this family of flow networks, and posted the sequence of maximal flows to OEIS: A238729. I can't find any reference to it either. Has anyone seen it?
Here is the description:
...

**-2**

votes

**1**answer

152 views

### Decimal digits multiplied by powers of 2: leads to mod 8? [closed]

This is more a puzzle than a research question,
a puzzle to me. Perhaps it is straightforward for others.
Imagine Repeatedly interpreting a number
expressed with the usual base-$10$ digits
as "digits"...

**2**

votes

**0**answers

61 views

### Regular graphs with unimodal subdegrees that are not distance-regular

Distance regular graphs are known to exhibit the following property: starting from an arbitrary vertex $\alpha$, let $k_i$ denote the number of vertices at distance $i$ from $\alpha$ (in terms of ...

**1**

vote

**1**answer

188 views

### A square-squareroot integer race sequence involving primes

I wonder what is the expected behavior of this process?
Let
$f^2_{\mathrm{next}}(n) =$ the next prime after $n^2$.
$g_{\mathrm{sqrt}}(n) = \lfloor \sqrt{n} \rfloor$.
Now iterate as follows,...