130 problems

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…

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

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…

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 $d(n)$ 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 $a_r$ 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…

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 $J_n(x)$ 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…

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 $\rho$ 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…

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

Conjectures for $L_n$ and $M_n$. The following five propositions are conjectured:

Let … mathfrak{S}{mu} denote the set of … mu in $\mathbb{Z}$, … The paper explains that 1\in\mathfrak{S}_{mu}$for every … mu-Sondow number other than$1…

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 $(a(n))_{n\geq 1}$ 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 defined by $a(1)=1$, $a(2)=2$, and, for $n\geq 3$, letting $a(n)$ be the smallest natural number not already in the sequence such that…

Let $G$ be Golomb's sequence, let $S(k)=\sum_{n=1}^{k}G(n)$, and let $N_r(n)$ denote the relevant multiplicity for order $r$; define $M(r)=\sup_n N_r(n)$. For each $k$, let $j_k$ b…

Let $d(n)$ be the Conway-type integer sequence, and let $I_K$ denote its generation domains, with $I_{-1}=\{1,2\}$. Power-of-two values conjecture. The sequence satisfies … and … T…

Let $(a_n)_{n\ge1}$ be the strictly increasing sequence defined by … and, for $k\ge3$, let $a_k$ be the least integer greater than $a_{k-1}$ that can be written as a sum of at leas…

Asymptotic growth conjecture. The golombic sequence $\gamma w$ is either bounded or