249 problems

Let $J_n(x)$ be the finite linear Jaco graph of order $n$, and let $\Delta(J_n(x))$ denote its maximum degree. A sequence is $p$-graphical for a graph family $\mathcal{F}$ when it…

Given a finite starting sequence of numbers, extend it by repeatedly appending the curling number of the sequence so far. The curling number is the largest $k$ for which the sequen…

Let $s=(s(n))_{n\geq 0}$ be a constant-recursive sequence. A general rank sequence is the rank sequence determined by a recurrence and its associated general exponential-polynomial…

Equality characterization conjecture. The upper bound is attained,

Golden-ratio formulas. For every $n\geq 1$,

For $x\in\mathbb{N}$ and $n\geq 3$, define … where $s_{n-1}(r)$ is the digit sum of $r$ in base $n-1$. Moser's conjecture. For every $x\in\mathbb{N}$, … Here $S_{3,\;0}(x)$ is the…

Let $A_{n,k}$ be the ballot numbers … For all $r\in\mathbb{N}$ and $n,s\in\mathbb{Z}^+$ such that $r+s\equiv 1\pmod 2$, and with $\varepsilon=\pm1$, Guo–Zeng's ballot-number divisi…

Let $J_n(x)$ be a finite linear Jaco graph and let $\Delta(J_n(x))_{\mathrm{unique}}$ denote a maximum degree that occurs uniquely in the relevant sequence of maximum degrees. A se…

Dekking's conjecture. For every integer $n\geq 1$,

Zagier's conjecture. Zagier's finite search yielded all the integral solutions to this recurrence.

Large-p prime fixed-point conjecture. For large $p$, the set of odd primes excluding $p$ can be obtained as fixed points of the sequence $A(p)$.

Farhi's conjecture. Every natural number can be represented as the sum of three terms of this sequence.

Curling number conjecture. For every initial sequence $S$, there exists $t\geq 0$ such that $\operatorname{cn}(S_t)=1$.

Let $m\geq4$, set $A_m=\{1,\ldots,m-3\}$, and define … with, for $n\geq2$, … Let $n^*=n^*(m)$ be the point of the last nontrivial increment of $\{\tilde{c}^{(m)}(n)\}$ on $A_m$, wi…

Let $\tilde{c}(n)$ be the sequence defined by … An increment is called main if it occurs at one of the distinguished fundamental points described in the paper. Shevelev's main-incr…

Let $T$ and $F$ be the transformations associated with codes, and let ${\mathbf c}_1,{\mathbf c}_2$ be codes of $T$ of equal length. For a code ${\mathbf c}$, let…

Let $f'(n)$ be the companion sequence described above, and let $g'(m)$ denote the position where $f'(n)$ first takes the value $m$. Let $g(m)$ denote the corresponding position for…

Smarandache cube-digital subsequence conjecture. All terms of the Smarandache cube-digital subsequence are of the form $D \times 10^{3k}$ where $D \in \left\{0,1,8\right\}$ and…

Approximate multiplication exceptional-set conjecture. The set $E(r)$ is finite. The expanding condition is necessary, and the paper proves the conjecture for rationals with denomi…

Let … be a linear recurrence which, for some initial values $n_1,n_2,\dots,n_r$, defines an infinite increasing sequence of natural numbers $S=\\{n_1,n_2,\dots\\}$. Let $\Phi_S(n)$…

Let $1\leq p\leq q\leq r$ and set $k=p+q+r$. Define a sequence by … with initial conditions $a_i=1$ for $0\leq i\leq k-1$. Robinson's integrality conjecture. The resulting sequence…

The sequences Somos-$4$ through Somos-$7$ are defined by the quadratic recurrences … with $n\geq4$ and $a_i=1$ for $0\leq i\leq3$; … with $n\geq5$ and $a_i=1$ for $0\leq i\leq4$; t…

Let $X$ be an indeterminate, let $n$ be a positive integer, and let $r=(r_1,\ldots,r_m)$ be a positive multi-integer with weight … For a partition $\mu$, write $l(\mu)$ for its len…

Let $(a(n))_{n\geq 1}$ be the EKG sequence defined by $a(1)=1$, $a(2)=2$, and, for $n\geq 3$, choosing the smallest unused natural number sharing a nontrivial common divisor with…

Let $(a(n))_{n\geq 1}$ be the EKG sequence, and let $p$ denote a prime. Write $f(n)\sim g(n)$ when $f(n)/g(n)\to 1$ as $n\to\infty$. The EKG asymptotic formula conjecture. If…