Related papers: Divisibility properties of Dedekind numbers

In this paper, we define the concept of the Study-type determinant, and we present some properties of these determinants. These properties lead to some properties of the Study determinant. The properties of the Study-type determinants are…

We study some divisibility properties of multiperfect numbers. Our main result is: if $N=p_1^{\alpha_1}... p_s^{\alpha_s} q_1^{2\beta_1}... q_t^{2\beta_t}$ with $\beta_1, ..., \beta_t$ in some finite set S satisfies…

We provide an algorithmic description of a family of graded decomposition numbers for rational Cherednik algebras.

Using a generalization due to Lerch [M. Lerch, Sur un th\'{e}or\`{e}me de Zolotarev. Bull. Intern. de l'Acad. Fran\c{c}ois Joseph 3 (1896), 34-37] of a classical lemma of Zolotarev, employed in Zolotarev's proof of the law of quadratic…

The divisibility of truncated binomial series by their exponent n is analyzed. Divisibility is shown to depends on the divisibility characteristics of the integers constituting the binomials. Series division by the highest possible powers…

The enumeration $d_k(n)$ of $k$-elongated plane partition diamonds has emerged as a generalization of the classical integer partition function $p(n)$. We have discovered an infinite congruence family for $d_5(n)$ modulo powers of 5.…

We study the properties of the odd Catalan numbers, C_n, modulo 2^k for k >= 2. We show that there exist only k - 1 different congruences of the odd Catalan numbers modulo 2^k. Moreover, these congruences can be obtained by C_{2^m - 1} (mod…

The partition function $ p_{[1^c11^d]}(n)$ can be defined using the generating function, \[\sum_{n=0}^{\infty}p_{[1^c{11}^d]}(n)q^n=\prod_{n=1}^{\infty}\dfrac{1}{(1-q^n)^c(1-q^{11 n})^d}.\] In this paper, we prove infinite families of…

We consider the number of the $6$-regular partitions of $n$, $b_6(n)$, and give infinite families of congruences modulo $3$ (in arithmetic progression) for $b_6(n)$. We also consider the number of the partitions of $n$ into distinct parts…

We define Dedekind sums attached to a totally real number field of class number one. We prove that they satisfy some reciprocity law. Then we relate them to special values of Hecke $L$-functions. We conclude that they are ruled by Stark's…

We show that each rational number $r$, $0\le r<1$, occurs as the fractional part of a Dedekind sum $S(m,n)$.

In this paper, we consider the set of partitions $pend(n)$ which enumerates the number of partitions of $n$ wherein the even parts are not allowed to be distinct. Using a result of Newman, we prove a few infinite families of congruences…

We classify the discriminantly separable polynomials of degree two in each of three variables, defined by a property that all the discriminants as polynomials of two variables are factorized as products of two polynomials of one variable…

We prove the congruence $\sum_{1 \leq k < \sqrt{N}} \sigma_0 (N - k^2) \equiv 0 \pmod 4$, where $\sigma_0(m)$ denotes the number of positive divisors of $m$, for $N = An + B$ with $(A,B) \in \{ (16,14),$ $(36,30),$ $(72,42),$ $(196,70),$…

Let $p_k(n)$ denote the number of $2$-color partitions of $n$ where one of the colors appears only in parts that are multiples of $k$. We will prove a conjecture of Ahmed, Baruah, and Dastidar on congruences modulo $5$ for $p_k(n)$.…

Let v(n) denote the number of compositions (ordered partitions) of a positive integer n into powers of 2. It appears that the function v(n) satisfies many congruences modulo 2^N. For example, for every integer B there exists (as k tends to…

We define Dedekind semidomains as semirings in which each nonzero fractional ideal is invertible. Then we find some equivalent condition for semirings to being Dedekind. For example, we prove that a Noetherian semidomain is Dedekind if and…

We prove that the moduli space of tetragonal curves of genus g>6 is rational when g is congruent to 1, 2, 5, 6, 9, 10 modulo 12 and not equal to 9, 45.

Let p(n, k) denote the number of partitions of n into parts less than or equal to k. We show several properties of this function modulo 2. First, we prove that for fixed positive integers k and m, p(n,k) is periodic modulo m. Using this, we…

We show that the necessary conditions for the existence of 4-GDDs of type g^u m^1 are sufficient for g congruent to 0 (mod h), h = 39, 51, 57, 69, 87, 93, and for g = 13, 17, 19, 23, 25, 29, 31 and 35. More generally, we show that for all g…