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Let the collection of arithmetic sequences ${d_{i} n + b_{i} : n \in ℤ}_{i \in I}$ be a disjoint covering system of the integers. We prove that if $d_{i} = p^{k} q^{l}$ for some primes $p, q$ and integers $k, l \geq 0$ , then there is a $j \neq i$ such that $d_{i} | d_{j}$ . We conjecture that the divisibility result holds for all moduli.A disjoint covering system is called saturated if the sum of the reciprocals of the moduli is equal to $1$ . The above conjecture holds for saturated systems with $d_{i}$ such that the product of its prime factors is at most $1254$ .

A density estimate for the $3 x + 1$ problem

Ivan Korec (1994)

Mathematica Slovaca

A few new facts about the EKG sequence.

Hofman, Piotr, Pilipczuk, Marcin (2008)

Journal of Integer Sequences [electronic only]

A function related to the central limit theorem

Paul Bracken (1998)

Commentationes Mathematicae Universitatis Carolinae

A number of properties of a function which originally appeared in a problem proposed by Ramanujan are presented. Several equivalent representations of the function are derived. These can be used to evaluate the function. A new derivation of an expansion in inverse powers of the argument of the function is obtained, as well as rational expressions for higher order coefficients.

A further note on a class of $I_{0}$ -sets

David Grow (1987)

Colloquium Mathematicae

A generalization of a theorem of Erdös on asymptotic basis of order $2$

Martin Helm (1994)

Journal de théorie des nombres de Bordeaux

Let $𝒯$ be a system of disjoint subsets of $ℕ^{*}$ . In this paper we examine the existence of an increasing sequence of natural numbers, $A$ , that is an asymptotic basis of all infinite elements $T_{j}$ of $𝒯$ simultaneously, satisfying certain conditions on the rate of growth of the number of representations $𝑟_{𝑛} (𝐴); 𝑟_{𝑛} (𝐴) : = |\{(𝑎_{𝑖}, 𝑎_{𝑗}) : 𝑎_{𝑖} < 𝑎_{𝑗}; 𝑎_{𝑖}, 𝑎_{𝑗} \in 𝐴; 𝑛 = 𝑎_{𝑖} + 𝑎_{𝑗}\}|$ , for all sufficiently large $n \in T_{j}$ and $j \in ℕ^{*}$ A theorem of P. Erdös is generalized.

A generalization of a theorem of Erdős-Rényi to m-fold sums and differences

Kathryn E. Hare, Shuntaro Yamagishi (2014)

Acta Arithmetica

Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define $r_{N}^{(m)} (ω)$ to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that $r_{N}^{(m)} (ω) < K$ for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.

A generalization of Beatty's theorem.

Holshouser, Arthur, Reiter, Harold (2001)

Southwest Journal of Pure and Applied Mathematics [electronic only]

A generalization of the question of Sierpiński on geometric progressions.

Luo, Jiagui, Yuan, Pingzhi (2010)

Journal of Integer Sequences [electronic only]

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