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

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While there are known examples of numbers expressible as a sum of two positive integer cubes in four distinct ways like $6963472309248 = 2421^3 + 19083^3 = 5436^3 + 18948^3 = 10200^3 + 18072^3 = 13320^...
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Let $s:\mathbb{N}\to\{0,1\}$ be the Champernowne sequence, starting with $$0\, 1\, 10\, 11\, 100\, 101\, 110\,\ldots$$ It is well known that this sequence is normal. Question. If $p:\mathbb{N}\to\...
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For any positive integer $n$, define $s(n)$ as the smallest positive integer $m$ such that the $n$ distinct numbers $$ (p_1-1)^2,\ (p_2-1)^2,\ \ldots,\ (p_n-1)^2$$ are pairwise incongruent modulo $m$,...
Zhi-Wei Sun's user avatar
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For any positive integer $n$, let $S(n)$ be the sum of the first $n$ primes. Then $$S(1) = 2,\ S(2)=2+3=5,\ S(3)=2+3+5 =10,\ S(4) = 2+ 3+5+7 =17.$$ By the Prime Number Theorem, $$S(n)\sim \frac{n^2}2\...
Zhi-Wei Sun's user avatar
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Consider the triangular array $T(n,k)_{1 \le k \le n}$ defined by the recurrence \begin{align*} T(n,1) &= 1, \\ T(n,k) &= 1+\sum_{i=1}^{k-1} T(n - i, k - 1) -\sum_{i=1}^{n-1} T(n - i, k). \end{...
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For $n\in\mathbb Z^+=\{1,2,3,\ldots\}$, let $p_n$ denote the $n$th prime. A well known conjecture of de Polignac states that for any $n\in\mathbb Z^+$ there are infinitely many $k\in\mathbb Z^+$ with $...
Zhi-Wei Sun's user avatar
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Let $n \ge 1$. A set of vectors $v_1, \ldots, v_m \in \{0,1\}^n$ is called admissible if all pairwise sums $v_i + v_j$ (with $1 \le i \le j \le m$) are distinct. We want to find the number $a(n)$, ...
Alexey Ustinov's user avatar
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I conjecture that For every integer $k>78$, there exists an odd prime $p$ such that the sum of last two base-$p$ digits of $k$ is $\geq p$. We may additionally assume that $k+1$ is a prime, a ...
Max Alekseyev's user avatar
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Let $a,b \in \mathbb{R}$ and sequece $\{f(n)\}_{n=1}^{\infty}$ is given by homogeneous second order recursive relation $$ f(n):=af(n-1)-b^2f(n-2), \:\:\: n>2 $$ with two arbitrary starting values $...
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Let $n$ be a positive integer, and let $\varphi$ be Euler's totient function. Clearly $n>\varphi(n)$ for any integer $n>1$. An open conjecture of Lehmer states that $$n\not\equiv1\pmod{\varphi(n)...
Zhi-Wei Sun's user avatar
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Let $a(n)$ be A000111, i.e., Euler or up/down numbers whose exponential generating function $f(x)$ satisfies $$ f(x) = \sec(x) + \tan(x). $$ $\varphi(n)$ be A000010, i.e., Euler totient function. ...
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The Domb numbers given by $$D_n:=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}k\binom{2(n-k)}{n-k}\ \ (n=0,1,2,\ldots)$$ arising from enumerative combinatorics have many interesting properties (cf. https://...
Zhi-Wei Sun's user avatar
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Consider the sequence defined as follows: Start with a number N. Compute the prime factors of N with multiplicity, and add 1. Then, sum this list together to get N'. Iterate this procedure until you ...
weissguy's user avatar
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Let $(a_n)_{n \ge 1}$ be a sequence of positive integers such that $$ a_1 + a_2 + \cdots + a_n \leq Kn \quad \text{for all } n \in \mathbb{N}, $$ for some constant $K > 0$. Is the following ...
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$\def\NN{\mathbb{N}^\mathbb{N}}$Let $\NN$ denote the set of sequences (maps) $f:\mathbb{N}\to\mathbb{N}$. Is there $u\in\NN$ such that for all $f\in\NN$ there is $s\in\NN$ such that $s$ is strictly ...
Dominic van der Zypen's user avatar
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Is $a(n)=(34^n+1)/(34+1)$ ever prime for odd $n>3$? $n$ must be prime because of polynomial factorization. There are no counterexamples up to $n=11,000$ and there are no congruence obstructions ...
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Let $a$ be an arbitrary non-decreasing integer sequence such that $a(1) = 2$. Here $a(n)$ is $n$-th term of $a$. $b$ be an integer sequence of numbers $k$ such that $a(k+1)>a(k)$. Here $b(n)$ is $...
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As in Question 491655 and Question 491762, we define $$a_n(x):=\sum_{i,j=0}^n\binom ni^2\binom nj^2\binom{i+j}ix^{i+j}$$ for each nonnegative integer $n$. Here we pose some curious congruences ...
Zhi-Wei Sun's user avatar
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Let $a(n)$ be an arbitrary integer sequence with $a(n)=0$ for $n \leqslant 0$ and ordinary generating function $A(x)$. $b(n)$ be an INVERT transform of $a(n)$ and whose ordinary generating function ...
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We call a finite subset $S\subseteq \mathbb{N}$ arithmetical if there are $n, k\in\mathbb{N}$ with $k>1$ such that $S = \{n+j: 0 \leq j\leq k\}$. Given an integer $\ell>0$ and a bijection $\...
Dominic van der Zypen's user avatar
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For a fixed $n$, the sequence is defined as $S_n(k+1) = S_n(k) \times S_n(k)$, with $S_n(0) = 2^n + 1$. It's quite obvious that the first $n$ binary digits of $S_n(n)$ match those of $e$, give or take....
Vincent Granville's user avatar
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In my research, I came across the following formula for Apéry numbers: $$ a_n = \sum_{i=0}^n \sum_{j=0}^n {n \choose i}^2 {n \choose j}^2{i+j \choose i}.$$ This formula does not seem to appear in the ...
Antoine Labelle's user avatar
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Let $f(n)$ be an arbitrary function with integer values. Let $T(n,k)$ be an integer coefficients such that $$ T(n,k) = \sum\limits_{j=2}^{n-k+1} f(j) \binom{-k}{j} T(n, k+j-1), \\ T(n,n) = 1. $$ I ...
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Let $a$ be a nonnegative integer not a multiple of $10$. Let $\nu_p(\ldots)$ indicate the $p$-adic valuation of the argument. We have recently introduced the so called "constant congruence speed ...
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Let $a(n,m)$ be the family of integer sequence such that $$ a(n,m) = \sum\limits_{i=0}^{n} \sum\limits_{j=0}^{i} \frac{(i+m)^{n-i+j}(-1)^{n-i}}{j!(n-i)!}. $$ I conjecture that for any $m$ we have $$ a(...
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Define the linear recurrence with polynomial coefficients $$a(0)=0,a(1)=1,a(n)=(n-1)a(n-1)+a(n-2)$$ It is A001040 and per comments is the numerator of the continued fraction $[1,2,3,\ldots,n-1]$ Let $...
joro's user avatar
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For natural $N$ define $\rho(N)$ to be the smallest positive solution of $x! \equiv (2x)! \pmod{N}$ $\rho(N) \le N$ since $N!$ vanishes. Q1 Are there any closed form properties of $\rho(N)$ known? Q2 ...
joro's user avatar
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Let $a(n)$ be A347420 (i.e., number of partitions of $[n]$ where the first $k$ elements are marked $(0 \leqslant k \leqslant n)$ and at least $k$ blocks contain their own index). Let $f(n,q)$ be an ...
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Question: What is the smallest odd integer $n>1$ such that $\phi(\tau(n))=\tau(\phi(n))$? OEIS A078148 has a list of the positive integers $n$ which satiafies $\phi(\tau(n))=\tau(\phi(n))$, where $\...
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Let $a(n,m)$ be the family of integer sequences such that $$ a(n,m) = (n+1)\left(\left[m + \sum\limits_{i=1}^{n+1} \frac{1}{i}\right](-1)^n + \sum\limits_{i=1}^{n} \binom{n+1}{i}\frac{1}{i}(i+1)^m(-1)^...
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Let $\newcommand{\tn}{\{0,1\}^\mathbb{N}}\tn$ be the collection of all infinite binary sequences. For $s\in\tn$ and $k\in\mathbb{N}$ let the left-shift of $s$ by $k$ positions, $\ell_k(s)\in \tn$, be ...
Dominic van der Zypen's user avatar
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We define $$S(n)=\sum_{a=2+(n\pmod 2)}^{n-2} \sharp(\{j,1\leq j<n \pmod{a},(a,j)=1\})\ .$$ (Searching the OEIS yielded no results.) For $n>2$ we have the following experimental observations (...
Roland Bacher's user avatar
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This is an integer sequence OEIS sequence A217703. It satisfies an order 3 recurrence which is the constant term $A_n=u_n(0)$ of a three term recurrent sequence of polynomial defined by $$u_0(x)=1,u_1(...
CHUAKS's user avatar
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Given $S \subseteq \mathbb{C}$, define $\displaystyle \mathfrak{c}(S) = \bigcap_{p(x) \in \mathbb{C}[x] \wedge p(S) \subseteq \mathbb{Z}}p^{-1}(\mathbb{Z}) \supseteq S$ ("the integral points ...
Zerox's user avatar
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Let $\omega$ denote the set of non-negative integers. For which integers $n>1$ is there a sequence $b_n: \omega\to\omega$ with the following property? Whenever $v\in\omega^n$ there is a unique $...
Dominic van der Zypen's user avatar
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10 votes
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Let $a_n$ be a sequence such that $a_1=1$ and for each $n \geq 1$ $a_{n+1}$ is the smallest positive integer distinct from $a_1,a_2,...,a_n$ such that $\gcd(a_{n+1}a_n+1,a_i)=1$ for each $i=1,2,...,n$....
jack's user avatar
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Here is a new conjecture of mine from the appendix of an unpublished manuscript currently under review. Let $b \in \mathbb{Z}^+$ and assume that $n$ is an integer greater than $1$ and not a multiple ...
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Let $C_n$ be A000108 (i.e., Catalan numbers). Here generating function is $C(x)$ such that $$ C(x) = \frac{1-\sqrt{1-4x}}{2x}. $$ Let $a(n)$ be an integer sequence with generating function $A(x)$ such ...
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I am interested in the distributedness or "mixing" behavior of certain linear recurrences modulo powers of $2$. In particular, consider the Pell sequence (https://oeis.org/A000129), modulo $...
gtm's user avatar
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I was recently looking at this problem: “There are a number of balls in a jar, some of them red, some of them white. The odds of picking two at random and both balls being red is 1/2. How many of the ...
Conor Pillay's user avatar
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For integers $n\ge k\ge0$, let $p(n,k)$ denote the number of ways to write $n$ as a sum of $k$ positive integers (repetition allowed). For example, $p(6,3)=3$ since $$6=1+1+4=1+2+3=2+2+2.$$ QUESTION. ...
Zhi-Wei Sun's user avatar
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Recently, I was studying prime sequences of the form $k \cdot 2^n + 1$, and I noticed that primes of the form $n \cdot 2^n + 1$ almost do not exist, except for the $n = 1$ case. Are there other prime ...
Arsen Vardanyan's user avatar
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A sequence of polynomials $$P_0(q),\ P_1(q),\ P_2(q),\ \ldots$$ with real coefficients is called $q$-log-convex if for each $n=1,2,3,\ldots$ every coefficient of the polynomial $P_{n+1}(q)P_{n-1}(q)-...
Zhi-Wei Sun's user avatar
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rowlands sequence is defined as follows \begin{equation} a_{n}=a_{n-1} + b_{n} \end{equation} where $b_{n} = gcd(a_{n-1}, n)$ for $n>h$ it originates from E. Rowlands 2008 paper "A Natural ...
Antisocialfreal's user avatar
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Let $T_q(n, k)$ be an integer table such that $$T_q(n, k) = \begin{cases} 1 & \textrm{if } n = 0 \vee k = 0 \\ qT_q(n-1, n-1) + T_q(n, n-1) & \textrm{if } n = k > 0 \\ T_q(n, k-1) + T_q(n-1,...
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For odd integer $n$ define the function $$ J(n)=(2^{\varphi(n)}-1) \bmod{n^2}$$ $J(n)$ is integer in $[0,n^2-1]$ and it is divisible by $n$. Integer $n$ is Wieferich number iff $J(n)=0$ and if $n$ is ...
joro's user avatar
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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
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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 $P$...
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2 votes
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By a distinct partition, I mean a partition into distinct parts, i.e., $10 = 5+4+1$ is one, but $10=6+2+2$ is not. The number of distinct partitions of $k$ all whose parts are at most $n$ is given by ...
Bubaya's user avatar
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We are interested which integer sequences are efficiently computable possibly over finite rings. Define the integer sequence $a(n)=(b_1 n +b_2)a(n-1)+(b_3 n + b_4)a(n-2)$ with initial terms $a(0),a(1)$...
joro's user avatar
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