130 problems

Let $r(x)$ be the generating function of the Rueppel sequence, and let the Hankel transform of a sequence be the sequence of determinants of its successive Hankel matrices. Periodi…

Let $(s_n)$ be an appropriate sequence of positive integers, and let $G_n$ be the action graph associated with $s_n$. For $n\geq 1$, write $z_n$ for the number of new vertices labe…

Let … and write its Stieltjes continued fraction as … Let $s_{b,c}(n)$ denote the corresponding Stieltjes parameters. Generalized Rueppel parameter conjecture. Then … where $P(k)$…

Bounded-gcd conjecture. For every Kurepa factorial sequence, $\mathcal{F}_n(a)$ is bounded above by $2$, with

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…

Central Stirling numbers conjecture. The binomial Ward numbers satisfy

For each $n\geq 2$, let $\mathcal{G}_n$ be the set of Gilbreath sequences of length $n$, let $K_S$ denote the valid-extension set associated with $S$, and let $U_n$ be the doubling…

Let $d(n)$ be the Conway-type integer sequence. For $K\geq -1$, let … with initial domain $I_{-1}=\{1,2\}$. For each $n$, let $n_1$ and $n_2$ be the two parent spots occurring in t…

Let $J_n(x)$ be a finite linear Jaco graph and let $X$ be its $\Delta$-set. Non-singleton-Delta-vertex conjecture. The vertex subscripts $i$ for which $v_i$ does not yield a single…

Let $J_n(x)$ be a finite linear Jaco graph of order $n\geq3$, and let $X$ be its $\Delta$-set, equivalently its Jaconian set, consisting of vertices attaining the maximum degree. A…

Let $r_n$ be the Rueppel sequence. Let $H_n$ be the Hankel transform of the sequence $1-r_n$, and let $h_n$ be the Hankel transform of the first-difference sequence $r_{n+1}-r_n$.…

Let a recurrence have characteristic polynomial with $k$ distinct roots, without multiplicity, and let $N\subseteq\mathbb{Z}^k$ be the subgroup of relations between these roots. Wr…

Let $\mathfrak{R}$ be the recursive map on positive integers defined by … where $\sigma$ is the sum-of-divisors function. Bounded descent conjecture. There exists a constant…

Write a positive integer $n$ in base $3$ as $n=d_1d_2\ldots d_m$, with digits $d_i\in\{0,1,2\}$, and let $cn(n)$ denote its comma-number. The notation $d_md_1$ denotes the two-digi…

Let $h_n$ be the Hankel transform of the Rueppel sequence $r_n$, and let $H_n$ be the Hankel transform of the once-shifted Rueppel sequence $r_{n+1}$. A268411 product conjecture. T…

Let $k\geq 9$, and let the $n$-th non-$k$-gonal pyramidal number be the $n$-th positive integer that is not a $k$-gonal pyramidal number. The preceding formula defines this sequenc…

Let $a_r$ be the order-$r$ almost Golomb sequence, defined by the sliding-window greedy recurrence in the source, and let $G$ denote Golomb's sequence. Prefix Conjecture. For all…

Fibonacci degree conjecture. Both polynomial expressions $\gamma_n(b_1^{m_1}\dots b_k^{m_k})$ and $\ell_n(b_1^{m_1}\dots b_k^{m_k})$ have degree $F_n$ in $m_j$ and degree $F_{n-1}$…

Let $\rho$ be the evaluation map used in the paper. Assuming the divisibility conjecture, for every integer $n$ define the square matrix $\mathbf{M}_n$ by … Let…

For a non-negative integer $n$, let $g(n)$ be the smallest integer $s\geq n$ for which there are distinct integers $a_1,\ldots,a_t\in[n,s]$, with $a_1=n$, such that…

Equality characterization conjecture. The upper bound is attained,

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 $a_r$ be the order-$r$ almost Golomb sequence, let $N_r(n)$ denote the multiplicity of the value $n$, and define $M(r)=\sup_n N_r(n)$. Domination Lemma. For all $r\ge 5$, … Tog…

Infinitude conjecture. For each positive integer $t\neq 2$, the set