Amicable pairs with few distinct prime factors

We investigate some equations involving the number of divisors d(n); the sum of divisors σ(n); Euler's totient function ϕ(n); the number of distinct prime factors ω(n); and the number of all prime factors (counted with multiplicity) Ω(n). The first part deals with equation f(xy) + f(xz) = f(yz). In the second part, as an analogy to x2 + y2 = z2, we study equation f(x2) + f(y2) = f(z2) and its generalization to higher degrees and more terms. We use just elementary methods and basic facts about the above functions and indicate why and how to discuss this topic in group study sessions or special maths classes of secondary schools in the framework of inquiry based learning.
 Subject Classification: 97F60, 11A25

For a positive integer [Formula: see text], if [Formula: see text] denotes the sum of the positive divisors of [Formula: see text], then [Formula: see text] is called a deficient perfect number if [Formula: see text] for some positive divisor [Formula: see text] of [Formula: see text]. In this paper, we prove some results about odd deficient perfect numbers with four distinct prime factors.

For a natural number \(n\), let \(A(n)\) denote the smallest natural number that has exactly \(n\) divisors. Let \(p_{1}p_{2}\ldots p_{k}\) be the prime decomposition of \(n\) where the primes are given in decreasing order and the factors are not necessarily distinct. If \(q_{1},\ldots ,q_{k}\) denote the first \(k\) primes and \(A(n)=q_{1}^{p_{1}-1}\ldots q_{k}^{p_{k}-1}\), we say that \(n\) is ordinary. If \(n\) is not ordinary, we say it is extraordinary. In Brown (2006), it was shown that almost all numbers are ordinary and if \(E_{x}\) denotes the set of extraordinary numbers less than or equal to a positive real number \(x\), \(|E_{x}|=o\left( \frac{x}{2^{(\log (\log x))^{\delta }}}\right) \) for any \(\displaystyle 0<\delta <1/2\). In what follows, we will prove that \(\displaystyle |E_x|=\frac{x}{\log x}e^{\Psi (x)}\) where \(\displaystyle \Psi (x)\sim \frac{1}{2\log 2}\frac{\log _{(3)}^2x}{\log _{(4)}x}\) and \(\displaystyle \log _{(r)}x\) denotes the \(r\)-times iterated natural logarithm. We make use of the Prime Number Theorem as well as estimates of the size of sets where the number of total prime factors or distinct prime factors is fixed.

Let be a positive integer and be the sum of the digits in basis q of the positive integer n. We prove that the quotient has a normal order one, where and are respectively, the number of distinct prime factors and the number of prime factors p of a positive integer n counted with multiplicity such that mod Moreover, we discuss sums of the form where f is a multiplicative function.

A weighted version of the Erdős–Kac theorem

Let \({q\geqq2}\) be an integer and denote S(n) the sum of the digits in base q of the positive integer n. Our main result is to estimate the sum \({\Sigma_{n\leqq x}\tilde{\omega}(n)}\) where \({\tilde{\omega}(n)}\) is the number of distinct prime factors p of n such that \({S(p) \equiv a \,{\rm mod} \,b \,(a \in \mathbb{Z}, b\geqq 2)}\).

On the normal number of prime factors of φ(n) subject to certain congruence conditions

Let [Formula: see text] be a positive integer, and define [Formula: see text] for [Formula: see text], where [Formula: see text] denotes the number of distinct prime factors of [Formula: see text], and [Formula: see text] represents the total number of prime factors of [Formula: see text] (counted with multiplicity). In this paper, we study these two zeta functions and related arithmetical functions. We show that [Formula: see text] which is similar to the known identity [Formula: see text] equivalent to the Prime Number Theorem. For [Formula: see text], we prove that [Formula: see text] We also raise a hypothesis on the parities of [Formula: see text] which implies the Riemann Hypothesis.

Let ℤ, ℕ, ℚ denote the sets of integers, positive integers and rational numbers, respectively. Solutions (x, y, m, n) of the equation (1) have been investigated in many papers:Let ω(m), ρ(m) denote the number of distinct prime factors and the greatest square free factor of m, respectively. In this note we prove the following results.

The set of the positive integers and positive square—free integers are denoted by IN and IN*, respectively, and we write IN(n) = IN ∩ [1, n], IN* (n) = IN* ∩ [1, n], where [1, n] = {1, 2, ... , n}. The set of primes is denoted by P. The smallest and greatest prime factors of the positive integer n are denoted by p(n) and P(n), respectively. ω(n) denotes the number of distinct prime factors of n, while Ω(n) denotes the number of prime factors of n counted with multiplicity: $$\omega \left( n \right)\, = \,\sum\limits_{p|n} {1,} \,\Omega \left( n \right)\, = \,\sum\limits_{{p^\alpha }\parallel n} {\alpha .} $$ µ(n) denotes the Mobius function.

The number of distinct prime factors of integers with missing digits is considered, and both the normal order and large values of the ω function over sets of this type are studied. A conjecture of Mauduit and Sarkozy, on large values of the Ω function over integers whose sum of digits is fixed, is also proved.

In this paper we use the Riemann zeta distribution to give a new proof of the Erdös-Kac Central Limit Theorem. That is, if ζ ( s ) = ∑ n ≥ 1 1 n s \zeta (s)=\sum _{n\ge 1} \frac {1}{n^s} , s &gt; 1 , s&gt;1, then we consider the random variable X s X_s with P ( X s = n ) = 1 ζ ( s ) n s , P(X_s=n)=\frac {1}{\zeta (s)n^s}, n ≥ 1. n\ge 1. In an earlier paper, the first author and Adrien Peltzer derived the analog of the Erdös-Kac Central Limit Theorem (CLT) for the number of distinct prime factors, ω ( X s ) , \omega (X_s), of X s , X_s, as s ↘ 1. s\searrow 1. In this paper we show, by means of a Tauberian Theorem, how to obtain the Central Limit Theorem of Erdös-Kac for the uniform distribution from the result for the random variable X s X_s . We also apply the technique to the number of distinct prime divisors of X s X_s that lie in an arithmetic sequence and a local CLT of the type proved by Dixit and Murty [Hardy-Ramanujan J. 43 (2020), 17–23] as well a version of the CLT for irreducible divisors of a monic polynomial over a finite field.

We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemerédi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements we can restrict the implicit parameter n n to those integers that have an even number of distinct prime factors or satisfy any other congruence condition. In order to obtain these refinements we study the limiting behavior of some closely related multiple ergodic averages with weights given by appropriately chosen multiplicative functions. These averages are then analyzed using a recent structural result for bounded multiplicative functions proved by the authors.

The energy of unitary cayley graphs

An odd perfect number, N N , is shown to have at least nine distinct prime factors. If 3 ∤ N 3\nmid N then N N must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect numbers.