arXiv subjects
Publications and source records attributed to Prapanpong Pongsriiam.
We introduce a new arc in directed graphs of integers. Among other things, we determine the positive integers that have arcs to all except a finite number of positive integers. We also propose some possible research problems at the end of this article.
For each $s\in \mathbb R$ and $n\in \mathbb N$, let $σ_s(n) = \sum_{d\mid n}d^s$. In this article, we give a comparison between $σ_s(an+b)$ and $σ_s(cn+d)$ where $a$, $b$, $c$, $d$, $s$ are fixed, the vectors $(a,b)$ and $(c,d)$ are linearly independent over $\mathbb Q$, and $n$ runs over all positive integers. For example, if $|s|\leq 1$, $a, b, c, d\in \mathbb N$ are fixed and satisfy certain natural conditions, then $$ σ_s(an+b) < σ_s(cn+d)\quad\text{ for all $n\leq M$} $$ where $M$ may be arbitrarily large, but in fact $σ_s(an+b) - σ_s(cn+d)$ has infinitely many sign changes. The results are entirely different when $|s|>1$, where the following three cases may occur: \begin{itemize} \item[(i)] $σ_s(an+b) < σ_s(cn+d)$ for all $n\in \mathbb N$; \item[(ii)] $σ_s(an+b) < σ_s(cn+d)$ for all $n\leq M$ and $σ_s(an+b) > σ_s(cn+d)$ for all $n\geq M+1$; \item[(iii)] $σ_s(an+b) - σ_s(cn+d)$ has infinitely many sign changes. \end{itemize} We also give several examples and propose some problems.
Let $n$ and $k$ be positive integers and $σ(n)$ the sum of all positive divisors of $n$. We call $n$ an exactly $k$-deficient-perfect number with deficient divisors $d_1, d_2, \ldots, d_k$ if $d_1, d_2, \ldots, d_k$ are distinct proper divisors of $n$ and $σ(n)=2n-(d_1+d_2+\ldots + d_k)$. In this article, we show that the only odd exactly $3$-deficient-perfect number with at most two distinct prime factors is $1521=3^2 \cdot 13^2$.
In this article, we give explicit formulas for the $p$-adic valuations of the Fibonomial coefficients ${p^a n \choose n}_F$ for all primes $p$ and positive integers $a$ and $n$. This is a continuation from our previous article extending some results in the literature, which deal only with $p = 2,3,5,7$ and $a = 1$. Then we use these formulas to characterize the positive integers $n$ such that ${pn \choose n}_F$ is divisible by $p$, where $p$ is any prime which is congruent to $\pm 2 \pmod{5}$.
A positive integer $n$ is said to be a palindrome in base $b$ (or $b$-adic palindrome) if the representation of $n = (a_k a_{k-1} \cdots a_0)_b$ in base $b$ with $a_k \neq 0$ has the symmetric property $a_{k-i} = a_i$ for every $i=0,1,2,\ldots ,k$. Let $s_b$ be the reciprocal sum of all $b$-adic palindromes. It is not difficult to show that $s_b$ converges. In this article, we obtain upper and lower bounds for $s_b$ and the inequality $s_{b} <s_{b'}$ for $2\leq b<b'$. Its consequences and some numerical data are also given.
We explicitly solve the diophantine equations of the form $$ A_{n_1}A_{n_2}\cdots A_{n_k}\pm 1 = B_m^2 $$ where $(A_n)_{n\geq 0}$ and $(B_m)_{m\geq 0}$ are either the Fibonacci sequence or Lucas sequence. This extends the result of D. Marques (2011) and L. Szalay (2012) concerning a variant of Brocard-Ramanujan equation. This is a manuscript of the article published in Communications of the Korean Mathematical Society, 32(3) (2017), pp. 511-522
Let $F_{n}$ and $L_n$ be the $n$th Fibonacci and Lucas number, respectively. For each positive integer $m$, the order of appearance of $m$ in the Fibonacci sequence, denoted by $z(m)$, is the smallest positive integer $k$ such that $m$ divides $F_k$. Recently, D. Marques has obtained a formula for $z(F_{n}F_{n+1})$, $z(F_{n}F_{n+1}F_{n+2})$, and $z(F_{n}F_{n+1}F_{n+2}F_{n+3})$. In this paper, we extend Marques' result to the case $z(F_{n}F_{n+1}\cdots F_{n+k})$ for every $4\leq k \leq 6$. We also give a formula for $z(L_nL_{n+1}\cdots L_{n+k})$ when $k = 5,6$ which extends the recent result of Marques and Trojovský. Our method gives a general idea on how to obtain the formulas for $z(F_nF_{n+1}\cdots F_{n+k})$ and $z(L_nL_{n+1}\cdots L_{n+k})$ for every $k\geq 1$.
We give the definition of uniform symmetric continuity for functions defined on a nonempty subset of the real line. Then we investigate the properties of uniformly symmetrically continuous functions and compare them with those of symmetrically continuous functions and uniformly continuous functions. We obtain some characterizations of uniformly symmetrically continuous functions. Several examples are also given.
Let $F_n$ be the $n$th Fibonacci number. Let $m, n$ be positive integers. Define a sequence $(G(k,n,m))_{k\geq 1}$ by $G(1,n,m) = F^m_n$, and $G(k+1,n,m) = F_{nG(k,n,m)}$ for all $k\geq 1$. We show that $F_n^{k+m-1}\mid G(k,n,m)$ for all $k, m, n\in\mathbb N$. Then we calculate $\frac{G(k,n,m)}{F_n^{k+m-1}}\pmod{F_n}$.
We extend the definition of weak symmetric continuity to be applicable for functions defined on any nonempty subset of $\R$. Then we investigate basic properties of weakly symmetrically continuous functions and compare them with those of symmetrically continuous functions and weakly continuous functions. Several examples are also given.
Kirk and Shahzad have recently given fixed point theorems concerning local radial contractions and metric transforms. In this article, we replace the metric transforms by metric-preserving functions. This in turn gives several extensions of the main results given by Kirk and Shahzad. Several examples are given. The fixed point sets of metric transforms and metric-preserving functions are also investigated.
Functions whose composition with every metric is a metric are said to be metric-preserving. In this article, we investigate a variation of the concept of metric-preserving functions where metrics are replaced by ultrametrics.
For a nonempty finite set $A$ of positive integers, let $\gcd\left(A\right)$ denote the greatest common divisor of the elements of $A$. Let $f\left(n\right)$ and $Φ\left(n\right)$ denote, respectively, the number of subsets $A$ of $\left\{1, 2, \ldots, n\right\}$ such that $\gcd\left(A\right) = 1$ and the number of subsets $A$ of $\left\{1, 2, \ldots, n\right\}$ such that $\gcd\left(A\cup\left\{n\right\}\right) =1$. Let $D\left(n\right)$ be the divisor sum of $f\left(n\right)$. In this article, we obtain partial sums of $f\left(n\right)$, $Φ\left(n\right)$ and $D\left(n\right)$. We also obtain a combinatorial interpretation and a congruence property of $D\left(n\right)$. We give open questions concerning $Φ\left(n\right)$ and $D\left(n\right)$ at the end of this article.
Four functions counting the number of subsets of $\{1, 2, ..., n\}$ having particular properties are defined by Nathanson and generalized by many authors. They derive explicit formulas for all four functions. In this paper, we point out that we need to compute only one of them as the others will follow as a consequence. Moreover, our method is simpler and leads to more general results than those in the literature.