8 problems

Let $\mathcal Q_H(x)$, $\mathcal Q_{H^-}(x)$, and $\mathcal Q_{H^+}(x)$ count the prime divisors up to $x$ of the $H$-, $H^-$-, and $H^+$-sequences, respectively. Let…

Let $\ell$ be a prime and let $1\leq a<d$ with $(a,d)=1$. A prime is $\ell$-Genocchi regular if it is not $\ell$-Genocchi irregular. Positive-density conjecture. If $\ell$ is odd,…

Let $\ell$ be an odd prime, let $1\leq a<d$ with $(a,d)=1$, and let $\mathcal P_G(d,a)(x)$ count the primes $p\leq x$ satisfying $p\equiv a\pmod d$ that are $\ell$-Genocchi irregul…

Let $n$ be a positive integer and define the $2n\times 2n$ matrix … The Luschny permanent conjecture. … Here $H_1,H_3,H_5,\ldots$ are the Genocchi numbers of the second kind, or me…

Lazar and Wachs' cycle-distribution conjecture. The number of D-permutations on $[2n]$ with $k$ cycles equals the number of E-permutations on $[2n]$ with $k$ cycles for all $k$. Co…

Even-odd-drop and D-cycle correspondence conjecture. For every finite subset $A\subset\mathbb{Z}_{>0}$, the number of cycles on $A$ with only even-odd drops is equal to the number…

Genocchi cycle-counting conjecture. For all $n\geq 1$, $g_n$ is equal to the number of cycles on $[2n]$ with only even-odd drops.

For each $ngeq 1$, let $\mathcal{D}_n$ be the $n$-chord diagram in which every chord intersects every other chord, and let $D_n=\varphi(\mathcal{D}_n)inmathbb{Z}[x]$. Write…