On sum of prime factors of composite positive integers

The Ramanujan Journal - Let $${\mathfrak {B}}(x)$$ be the number of composite positive integers up to x whose sum of distinct prime factors is a prime number. Luca and Moodley proved that there...     

On the density of sequences of integers the sum of no two of which is a square II. General sequences

The aim of this paper is to study the maximal density attainable by a sequence S of positive integers having the property that the sum of any two distinct elements of S is never a square. It is shown that there is a constant N₀ such that for all N ? N₀ any set S ? [1, N] having this property must have |S| < 0.475N. The proof uses the Hardy-Littlewood circle method.     

On binary signed digit representations of integers

Applications of signed digit representations of an integer include computer arithmetic, cryptography, and digital signal processing. An integer of length n bits can have several binary signed digit (BSD) representations and their number depends on its value and varies with its length. In this paper, we present an algorithm that calculates the exact number of BSDR of an integer of a certain length. We formulate the integer that has the maximum number of BSDR among all integers of the same length. We also present an algorithm to generate a random BSD representation for an integer starting from the most significant end and its modified version which generates all possible BSDR. We show how the number of BSD representations of k increases as we prepend 0s to its binary representation.     

On the number of primitive representations of integers as sums of squares

Formulas for the number of primitive representations of any integer n as a sum of k squares are given, for 2 ≤ k ≤ 8, and for certain values of n, for 9 ≤ k ≤ 12. The formulas have a similar structure and are striking for their simplicity. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—11E25; Secondary—05A15, 33E05.     

On a conjecture for representations of integers as sums of squares and double shuffle relations

In this paper, we prove a conjecture of Chan and Chua for the number of representations of integers as sums of $8s$ integral squares. The proof uses a theorem of Imamo?lu and Kohnen, and the double shuffle relations satisfied by the double Eisenstein series of level 2.     

On optimal binary signed digit representations of integers

Binary signed digit representation （BSD-R） of an integer is widely used in computer arithmetic, cryptography and digital signal processing. This paper studies what the exact number of optimal BSD-R of an integer is and how to generate them entirely. We also show which kinds of integers have the maximum number of optimal BSD-Rs.     

On cyclic compositions of positive integers

Say that two compositions of n into k parts are related if they differ only by a cyclic shift. This defines an equivalence relation on the set of such compositions. Let ${\left\langle \begin{array}{c}n \\ k\end{array} \right\rangle}$ denote the number of distinct corresponding equivalence classes, that is, the number of cyclic compositions of n into k parts. We show that the sequence ${\left\langle\begin{array}{c}n \\ k\end{array}\right\rangle}$ is log-concave and prove some results concerning ${\left\langle \begin{array}{c}n \\ k \end{array} \right\rangle}$ modulo two.     

Additive representations of integers

In this paper we shall mainly study additive representations of integers prime to the firstm primes as a sum of some integers having a peculiar property. The conjectures of Goldbach and twin primes are also observed in connection with these representations of integers.     

Five consecutive positive odd numbers none of which can be expressed as a sum of two prime powers II

In this paper, we find two integers k0, m of 159 decimal digits such that if k ≡ k0 （mod m）, then none of five consecutive odd numbers k, k - 2, k - 4, k - 6 and k - 8 can be expressed in the form 2^n ± p^α, where p is a prime and n, α are nonnegative integers.     

Multiplicative representations of integers

Lehh ≧ 2, and let ?=(B ₁, …,B _h ), whereB ₁ ? N={1, 2, 3, …} fori=1, …,h. Denote by g_?(n) the number of representations ofn in the formn=b ₁ …b _h , whereb _i ∈B _i . If v (n) > 0 for alln >n ₀, then ? is anasymptotic multiplicative system of order h. The setB is anasymptotic multiplicative basis of order h ifn=b ₁ …b _n is solvable withb _i ∈B for alln >n ₀. Denote byg(n) the number of such representations ofn. LetM(h) be the set of all pairs (s, t), wheres=lim g_? (n) andt=lim g_? (n) for some multiplicative system ? of orderh. It is proved that {fx129-1} In particular, it follows thats ≧ 2 impliest=∞. A corollary is a theorem of Erdös that ifB is a multiplicative basis of orderh ≧ 2, then lim g_? g(n)=∞. Similar results are obtained for asymptotic union bases of finite subsets of N and for asymptotic least common multiple bases of integers.     

Additive representations of integers

In this paper we shall mainly study additive representations of integers prime to the first m primes as a sum of some integers having a peculiar property. The conjectures of Goldbach and twin primes are also observed in connection with these representations of integers. Received: 20 October 1997     

On the sum of digits of some sequences of integers

Let b ≥ 2 be a fixed positive integer. We show for a wide variety of sequences {a _n } _n=1 ^∞ that for almost all n the sum of digits of a _n in base b is at least c _b log n, where c _b is a constant depending on b and on the sequence. Our approach covers several integer sequences arising from number theory and combinatorics.     

On the probability that k positive integers are relatively prime II

Given a set S of positive integers let $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t)}$ denote the number of k-tuples 〈m₁, …, m_k〉 for which $\text{m}{}_{\mathrm{i}}\text{∈ S ? [1, t]}$ and (m₁, …, m_k) = 1. Also let $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n)}$ denote the probability that k integers, chosen at random from $\text{S ? [1, n]}$, are relatively prime. It is shown that if P = {p₁, …, p_r} is a finite set of primes and S = {m : (m, p₁ … p_r) = 1}, then $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t) = (td(S))}{}^{\mathrm{k}}\text{Π}{}_{\mathrm{\nu ?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) + O(t}{}^{\mathrm{k?1}}\text{)}$ if k ≥ 3 and $\text{Z}{}_2{}^{\mathrm{S}}\text{(t) = (td(S))}{}^2\text{Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^2\text{) + O(t}\text{log}\text{t)}$ where d(S) denotes the natural density of S. From this result it follows immediately that $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) = (ζ(k))}{}^{\mathrm{?1}}\text{Π}{}_{\mathrm{p\in P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{)}{}^{\mathrm{?1}}$ as n → ∞. This result generalizes an earlier result of the author's where $\text{P = ?}$ and S is then the whole set of positive integers. It is also shown that if S = {p₁^x₁ … p_r^x_r : x_i = 0, 1, 2,…}, then $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → 0}$ as n → ∞.     

On the distribution of digits in Cantor representations of integers

Asymptotic formulas on the average values of the “sum of digits” function and the average numbers of occurrences of fixed subblocks in Cantor representations of integers are established. The theorems generalize a result by H. Delange [Enseign. Math.21 (1975), 31–47].     

On the structure of a random sum-free set of positive integers

Cameron introduced a natural probability measure on the set of sum-free sets, and asked which sets of sum-free sets have a positive probability of occurring in this probability measure. He showed that the set of subsets of the odd numbers has a positive probability, and that the set of subsets of any sum-free set corresponding to a complete modular sum-free set also has a positive probability of occurring. In this paper we consider, for every sum-free set S, the representation function r_s(n), and show that if r_s(n) grows sufficiently quickly then the set of subsets of S has positive probability, and conversely, that if r_s(n) has a sub-sequence with suitably slow growth, then the set of subsets of S has probability zero. The results include those of Cameron mentioned above as particular cases.