Skip to main content

All Questions

Filter by
Sorted by
Tagged with
6 votes
1 answer
367 views

On A057985 and A287066

Let $a(n)$ be A057985 (i.e., start with $0$ and repeatedly substitute: $0 \to 01$, $1 \to 12$, $2 \to 0$). Let $b(n)$ be A287066 (i.e., start with $1$ and repeatedly substitute: $0 \to 01$, $1 \to 12$...
Notamathematician's user avatar
6 votes
1 answer
282 views

Integer sequences with a periodic pattern

Let $A$ and $B$ be two different integers. Let $S$ be a finite integer sequence with exactly $n_A$ $A$s and $n_B$ $B$s. By repeating $S$ infinitely many times we obtain an infinite integer sequence $P$...
De Costa's user avatar
6 votes
1 answer
393 views

Test for pair of odd primes $(p, 2p^2-1)$

Let $a(n)$ be A106483 (i.e., primes $p$ such that $2p^2-1$ is also prime). Let $b(n)$ be an integer sequence such that $b(n) = B$ after the whole transformation where we start with $A = n$, $B = 1$, $...
Notamathematician's user avatar
6 votes
1 answer
402 views

Values of the determinants $\det[(j-k)^m+\delta_{jk}]_{1\le j,k\le n}\ (m=1,2,3,\ldots)$

For positive integers $m$ and $n$, let $D_m(n)$ denote the determinant $\det[(j-k)^m+\delta_{jk}]_{1\le j,k\le n}$, where the Kronecker delta $\delta_{jk}$ is $1$ or $0$ according as $j=k$ or not. ...
Zhi-Wei Sun's user avatar
  • 15.6k
6 votes
1 answer
315 views

Integral points of polynomials - a Furstenburg-type "topology" on $\mathbb{Z}$

Given $S \subseteq \mathbb{C}$, define $\displaystyle \mathfrak{c}(S) = \bigcap_{p(x) \in \mathbb{C}[x] \wedge p(S) \subseteq \mathbb{Z}}p^{-1}(\mathbb{Z}) \supseteq S$ ("the integral points ...
Zerox's user avatar
  • 1,543
6 votes
2 answers
389 views

Conjectured Somos-like closed form of recurrences with polynomial coefficients

From Our short paper For polynomial $F$ with integer coefficients, define the recurrence $f(n)=F(n,f(n-1),f(n-2),...,f(n-d))$. We conjecture that $f(n)$ satisfy Somos like sequence $f(n)=\frac{G(f(n-1)...
joro's user avatar
  • 25.4k
6 votes
1 answer
268 views

Sequence that sums up to the number of permutations avoiding the pattern $1-23-4$

Let $a(n)$ be A113227, i.e., the number of permutations on $[n]\equiv \{1, \ldots, n\}$ avoiding the pattern $1-23-4$. The sequence begins with $$1, 1, 2, 6, 23, 105, 549, 3207, 20577, 143239, 1071704,...
Notamathematician's user avatar
6 votes
1 answer
438 views

The largest digital sum of the square of an n-digit number

The sequence $13, 31, 46, 63, 81, 97, 112, 130, 148, 162, 180, \dots,$ (sequence A348300 in the OEIS) shows the largest digital sum the square of an $n$-digit (decimal) number has. Is this sequence ...
Bernardo Recamán Santos's user avatar
6 votes
1 answer
240 views

On the growth and bounds for a certain sequence of integers known as Bogotá numbers

A Bogotá number is a non-negative integer equal to some smaller number, or itself, times its digital product, i.e. the product of its digits. For example, 138 is a Bogotá number because 138 = 23 x (2 ...
Bernardo Recamán Santos's user avatar
6 votes
0 answers
171 views

An inequality involving integer partitions

For integers $n\ge k\ge0$, let $p(n,k)$ denote the number of ways to write $n$ as a sum of $k$ positive integers (repetition allowed). For example, $p(6,3)=3$ since $$6=1+1+4=1+2+3=2+2+2.$$ QUESTION. ...
Zhi-Wei Sun's user avatar
  • 15.6k
6 votes
0 answers
245 views

Searching for a proof of the pattern and identification of integer coefficients for the A329369

Please see the update given below. Everything you need to know from the old version of the question are the functions $a(n), \ell(n), s(n), t(n), r(n)$. Let $a(n)$ be A329369 (i.e, number of ...
Notamathematician's user avatar
6 votes
0 answers
286 views

Does $a_{i}(n)$ hit every positive integers infinitely many times for all $i\ge1$?

This question is related to a family of sequences. I have a simple definition as below and I have a question based on my limited observations for $i\le200$ and $n \le 10^{9}$. Definition. $a_{i}(1) = ...
Alkan's user avatar
  • 701
6 votes
0 answers
284 views

Is there a positive odd $n$ such that $\sigma(\sigma(n)) = \sigma(\sigma(n)-n)+\sigma(n)$?

Let $\sigma(n)$ denote the sum of the divisors of $n$. (https://oeis.org/A000203) It is relatively easy to find numbers $n$ such that $f(g(n)) = g(f(n))$ where $f(n) = \sigma(n)$ and $g(n) = \sigma(n) ...
Alkan's user avatar
  • 701
6 votes
0 answers
207 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(-e^...
joro's user avatar
  • 25.4k
5 votes
5 answers
677 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 ...
joro's user avatar
  • 25.4k
5 votes
2 answers
393 views

What is this sequence counting?

While solving (a system of) a system of linear equations level-by-level recursively, I am finding some redundant equations for level $n\geq5$. The reason why the redundancies arise is because $P(n)\...
TheTwistedSector's user avatar
5 votes
1 answer
172 views

On vanishing of $p$-adic logarithms

Might be related to Wieferich primes. Let $p$ be odd prime and define the Fermat quotient $$F(n)=\frac{(2^{n-1} -1)}{n} \mod n=\frac{(2^{n-1} \bmod n^2 )-1}{n}$$ For integer $b$ let $L_p(b)$ be the $p$...
joro's user avatar
  • 25.4k
5 votes
2 answers
1k views

Proof that $3^ns + \sum_{k=0}^{n-1} 3^{n-k-1}2^{a_k}=2^m.$

How would I go about proving the following: For any odd positive integer $s$, there exists a sequence of nonnegative integers $( a_0, a_1, \cdots, a_{n-1})$ and a nonnegative integer $m$ such that, $...
ReverseFlowControl's user avatar
5 votes
1 answer
297 views

Additive basis of order 2

Can we find $\alpha>1$ such that $u=(\lfloor n^\alpha\rfloor)_{n\geqslant0}$ is an additive basis of order $2$ (i.e. $\forall x\in\mathbb{N}, \exists(n,m)\in\mathbb{N}^2, x=u_n+u_m$) ? Remark : ...
uvdose's user avatar
  • 655
5 votes
1 answer
303 views

Simply generated sequences with mysterious differences

Suppose that $a_0 < a_1,$ $b_0 < b_1,$ and $$a_n=a_1b_{n-1}+a_0b_{n-2}+qn+r$$ for $n \geq 2$, where $a_0,a_1,b_0,b_1,q,r$ are integers such that $(a_n)$ and $(b_n)$ are increasing and ${(|a_n|)}$...
Clark Kimberling's user avatar
5 votes
1 answer
310 views

In the Oldenburger-Kolakoski sequence, is #1s = #2s infinitely many times?

The Oldenburger-Kolakoski sequence, $OK$, is the unique sequence of $1$s and $2$s that starts with $1$ and is its own runlength sequence: $$OK = (1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,\ldots).$...
Clark Kimberling's user avatar
5 votes
1 answer
184 views

Integer sequences with unique $k$-subsets sum

let the $\binom{\mathfrak{M}}{k}$ be a shorthand notation for chosing $k$ elements of set $\mathfrak{M}$ of positive integers and let $\left|\binom{\mathfrak{M}}{k}\right|$ denote the sum of the ...
Manfred Weis's user avatar
  • 13.2k
5 votes
1 answer
359 views

Discrete logarithm and the sequence $a(n)=(g^n \bmod p)^{p-1} \bmod p^2$

Let $p$ be prime and $g,n$ integers. Define $a(n)=(g^n \bmod p)^{p-1} \bmod p^2$ By mod p we don't mean congruence, but the reduction modulo $p$ operator. $A \bmod ...
joro's user avatar
  • 25.4k
5 votes
0 answers
307 views

On $s$-additive sequences

For a non-negative integer $s$, a strictly increasing sequence of positive integers $\{a_n\}$ is called $s$-additive if for $n>2s$, $a_n$ is the least integer exceeding $a_{n-1}$ which has ...
Sayan Dutta's user avatar
5 votes
0 answers
133 views

Formula and smallest solution for the A260711

Let $a(n)$ be A260711 without initial $0$ (i.e., numbers of the form $x^2 - y^2$ with $x > y$ where $x$ and $y$ are odd, $x + y$ is a power of $2$). The sequence begins with $$ 8, 16, 32, 48, 64, ...
Notamathematician's user avatar
5 votes
0 answers
161 views

Consecutive integers each of which has a large prime factor

There are many results about consecutive integers all having small prime factors. But what about consecutive integers each of which has a large prime factor? More precisely, let $P(n)$ be the ...
Penchez's user avatar
  • 341
5 votes
0 answers
317 views

Elliptic curve sequences needed for universal forgery

Elliptic Curve Digital Signature Algorithm (ECDSA) admits universal forgery (UF) if the Attacker can solve the equation $$z=\frac{f_{k-1}(x,y)f_{k+1}(x,y)}{f_{k}(x,y)^2},$$ where $k$ is unknown, $f_{k}...
Alexey Ustinov's user avatar
5 votes
0 answers
176 views

Can the integers in an easily computable sequence free of prime numbers always be factored easily?

Call a sequence $(a_n)$ of positive integers easily computable if there is a constant $C$ and an algorithm which computes $a_n$ from $n$, $a_1, \dots, a_{n-1}$ and a finite number of integer ...
Stefan Kohl's user avatar
  • 19.6k
4 votes
1 answer
241 views

Do there exist prime numbers of the form $n \cdot 2^n + 1$, when $n \in \mathbb{N}$ and $n > 1$?

Recently, I was studying prime sequences of the form $k \cdot 2^n + 1$, and I noticed that primes of the form $n \cdot 2^n + 1$ almost do not exist, except for the $n = 1$ case. Are there other prime ...
Arsen Vardanyan's user avatar
4 votes
2 answers
2k 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 $n=\mathrm {lcm} (1,2 \dots k)$, $G(n)$ ...
joro's user avatar
  • 25.4k
4 votes
2 answers
594 views

Squares in Lucas sequences

Good night, everyone! According to a celebrated result by J. H. Cohn, the only perfect squares in the Fibonacci sequence are $F_{0}=0$, $F_{1}=F_{2}=1$, and $F_{12}=144$. It is also known that the ...
Jamai-Con's user avatar
4 votes
1 answer
2k views

What do we know about Lucky numbers?

I'm really fascinated by lucky numbers (Wikipedia; OEIS A000959) and their prime-like characteristics. Wolfram states: write "out all odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The ...
Happydugongo's user avatar
4 votes
1 answer
168 views

An inequality involving $k$-generalized Fibonacci numbers

I have worked on a Diophantine equation by using transcendental and reduction methods given by Baker and Davenport. However, to solve completely the equation I have one complicated case and I proved ...
Davi's user avatar
  • 41
4 votes
1 answer
304 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)^{(n-...
joro's user avatar
  • 25.4k
4 votes
1 answer
148 views

Closed form for the A110501 (unsigned Genocchi numbers (of first kind) of even index)

Let $a(n)$ be A110501 (i.e., unsigned Genocchi numbers (of first kind) of even index). Here $$ a(n) = \sum\limits_{i=1}^{\left\lfloor\frac{n}{2}\right\rfloor}\binom{n}{2i}a(n-i)(-1)^{i-1}, \\ a(1) = 1 ...
Notamathematician's user avatar
4 votes
2 answers
303 views

Periods of natural numbers

Define a function $F$ on the natural numbers $\geq 2$ as follows: Start with $a \geq 2$ and let $b$ be the smallest prime divisor of $a$ and $c:=a+b$ and let $d$ denote the largest prime divisor of $c$...
Mare's user avatar
  • 26.5k
4 votes
1 answer
219 views

Numbers $n$ whose representation as the product of two divisors require more digits than that of $n$

Note: Posting in MO since it was unanswered in MSE Let $f(x)$ be the number of digits in the decimal representation of $x$ e.g. $, f(0) = 1, f(1729) = 4$. If $n = ab$ then we can show that $f(ab) > ...
Nilotpal Kanti Sinha's user avatar
4 votes
1 answer
217 views

Why do convoluted convolved Fibonacci numbers pop up from this triangle?

Start with this triangle (OEIS A118981). This triangle is simple to generate with the following recurrence relation (though $T(0,0)$ ends up different from the OEIS version): $$ T(0,0) = 2;T(1,0) = 1;...
Mitch's user avatar
  • 194
4 votes
1 answer
435 views

Quadratic progressions with very high prime density

In my previous MO question (see here), I solved the case for arithmetic progressions $f_k(x)=q_k x+1$. The solution is this: The list of sequences $f_k(x)$, each one corresponding to a specific $k$, ...
Vincent Granville's user avatar
4 votes
0 answers
121 views

Do all nonnegative integers appear in A051521?

For every positive integer $n$, $\tau(n)$ is the number of divisors of $n$. If we list the ratio of each positive integer $n$ to $\tau(n)$,they form a rational sequence 1,1,3/2,4/3,5/2,3/2,… Because $\...
Tong Lingling's user avatar
4 votes
0 answers
306 views

How to explain this number-theoretic seeming “almost coincidence”?

For natural numbers $n\geq2$, let $d(n)$ be the number of divisors of $n$, and let \begin{equation} g(n)=n\sum_i r_i(p_i-1) \end{equation} where $n=\prod_i p_i^{r_i}$ is the factorisation of $n$ as a ...
Simon's user avatar
  • 141
4 votes
0 answers
414 views

Explicit formula for tournament sequence

I am looking for an explicit formula for a sequence. The sequence is generated as follows: There is a tournament with $10$ teams. In the beginning, all teams have a 0-0 win-loss record. The teams are ...
Jackson's user avatar
  • 41
4 votes
0 answers
300 views

On $\sum_{k=1}^nk^3 = x^3 + y^3$ with $x,y \ge 1$

My question is related to https://oeis.org/A269839. It is well-known that there are parametric families of solutions for cubes that are sums of consecutive cubes: https://arxiv.org/pdf/1603.08901.pdf. ...
Alkan's user avatar
  • 701
4 votes
0 answers
97 views

When is $\lfloor C^n \rfloor \mod b$ efficiently computable?

For real irrational $C > 1 $ and natural $n,b$, define $a(C,n,b)=\lfloor C^n \rfloor \mod b$ Q1 For which $C,b$ is $a(C,n,b)$ computable in time polynomial in $\log{n}$? Searching in OEIS ...
joro's user avatar
  • 25.4k
4 votes
0 answers
178 views

Primitive roots modulo primes related to Fibonacci numbers or Lucas numbers

The Fibonacci numbers $F_0,F_1,F_2,\ldots$ and the Lucas numbers $L_0,L_1,L_2,\ldots$ are given by $$F_0=0,\ F_1=1,\ \text{and}\ F_{n+1}=F_n+F_{n-1}\ (n=1,2,3,\ldots)$$ and $$L_0=2,\ L_1=1,\ \text{...
Zhi-Wei Sun's user avatar
  • 15.6k
4 votes
0 answers
206 views

Generating a Penrose tessellation around a given tile

Given a starting Penrose tile, I need to build a "spiraling" tessellation around it. The following picture illustrates the request: In this example, the starting tile is a "thin rhombus" (the pink ...
Andrea Prunotto's user avatar
3 votes
1 answer
435 views

What is the connection between these three methods of generating this sequence?

I was recently looking at this problem: “There are a number of balls in a jar, some of them red, some of them white. The odds of picking two at random and both balls being red is 1/2. How many of the ...
Conor Pillay's user avatar
3 votes
1 answer
233 views

Min problem on integers

Let $n$ be any integer greater than $2^{10^6}$. Given any $s\le (\log_2 n)/1000$ integers $1=q_1\le q_2\le \cdots q_{s-1}\le q_s=n$. Prove that $$\min_\ell\left(\sum_{i=1}^\ell q_i\right)\left(\sum_{i=...
Nader Bshouty's user avatar
3 votes
1 answer
165 views

Are there infinitely many nonzero Euler quotients $a(n)=\frac{2^{\phi(n)}-1}{n} \bmod n$?

This might be related to an open problem. For odd natural $n$ define the Euler quotient: $$ a(n)=\frac{(2^{\phi(n)}-1) \bmod n^2}{n}=\frac{2^{\phi(n)}-1}{n} \bmod n$$ Q1 Are there infinitely many $n$ ...
joro's user avatar
  • 25.4k
3 votes
3 answers
696 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 $x$ ...
Larry Freeman's user avatar