Remainder sum and quotient sum function

2015 ◽  
Vol 07 (01) ◽  
pp. 1550001
Author(s):  
A. David Christopher
Keyword(s):  
Positive Integer ◽  
Periodic Function ◽  
Integer Partition ◽  
Integer Sequences ◽  
Partition Theory ◽  
Arithmetical Functions ◽  
Finite Set ◽  
Online Encyclopedia ◽  
Divisibility Properties ◽  
Positive Integers

This paper is concerned with two arithmetical functions namely remainder sum function and quotient sum function which are respectively the sequences A004125 and A006218 in Online Encyclopedia of Integer Sequences. The remainder sum function is defined by [Formula: see text] for every positive integer n, and quotient sum function is defined by [Formula: see text] where q(n, i) is the quotient obtained when n is divided by i. We establish few divisibility properties these functions enjoy and we found their bounds. Furthermore, we define restricted remainder sum function by RA(n) = ∑k∈A n mod k where A is a set of positive integers and we define restricted quotient sum function by QA(n) = ∑k∈A q(n, k). The function QA(n) is found to be a quasi-polynomial of degree one when A is a finite set of positive integers and RA(n) is found to be a periodic function with period ∏a∈A a. Finally, the above defined four functions found to have recurrence relation whose derivation requires few results from integer partition theory.

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

scholarly journals On the 2-adic order of Stirling numbers of the second kind and their differences

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Binary Representation ◽  
Exact Order ◽  
Special Cases ◽  
International Audience ◽  
The Difference ◽  
Divisibility Properties ◽  
Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

scholarly journals ON A VARIATION OF A CONGRUENCE OF SUBBARAO

2012 ◽  
Vol 93 (1-2) ◽  
pp. 85-90 ◽  
Author(s):  
ANDREJ DUJELLA ◽  
FLORIAN LUCA
Keyword(s):  
Positive Integer ◽  
Continued Fractions ◽  
Euler Function ◽  
Prime Factors ◽  
Finite Set ◽  
Positive Integers ◽  
Sum Of Divisors

AbstractWe study positive integers $n$ such that $n\phi (n)\equiv 2\hspace{0.167em} {\rm mod}\hspace{0.167em} \sigma (n)$, where $\phi (n)$ and $\sigma (n)$ are the Euler function and the sum of divisors function of the positive integer $n$, respectively. We give a general ineffective result showing that there are only finitely many such $n$ whose prime factors belong to a fixed finite set. When this finite set consists only of the two primes $2$ and $3$ we use continued fractions to find all such positive integers $n$.

scholarly journals Equisum Partitions of Sets of Positive Integers

Algorithms ◽  
2019 ◽  
Vol 12 (8) ◽  
pp. 164
Author(s):  
Eggleton
Keyword(s):  
Positive Integer ◽  
Initial Interval ◽  
Finite Set ◽  
Positive Integers

Let V be a finite set of positive integers with sum equal to a multiple of the integer b. When does V have a partition into b parts so that all parts have equal sums? We develop algorithmic constructions which yield positive, albeit incomplete, answers for the following classes of set V, where n is a given positive integer: (1) an initial interval a∈Z+:a≤n; (2) an initial interval of primes p∈P:p≤n, where P is the set of primes; (3) a divisor set d∈Z+:d|n; (4) an aliquot set d∈Z+:d|n, d<n. Open general questions and conjectures are included for each of these classes.

scholarly journals FACTORS OF SUMS AND ALTERNATING SUMS INVOLVING BINOMIAL COEFFICIENTS AND POWERS OF INTEGERS

2011 ◽  
Vol 07 (07) ◽  
pp. 1959-1976 ◽  
Author(s):  
VICTOR J. W. GUO ◽  
JIANG ZENG
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Binomial Coefficients ◽  
Alternating Sums ◽  
Divisibility Properties ◽  
Positive Integers

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers n1,…,nm, nm+1 = n1, and any nonnegative integer r, there holds [Formula: see text] and conjecture that for any nonnegative integer r and positive integer s such that r + s is odd, [Formula: see text] where ε = ±1.

PARTITIONS WITH PARTS IN A FINITE SET

2006 ◽  
Vol 02 (03) ◽  
pp. 455-468 ◽  
Author(s):  
ØYSTEIN J. RØDSETH ◽  
JAMES A. SELLERS
Keyword(s):  
Partition Function ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Residue Class ◽  
Finite Set ◽  
Positive Integers

For a finite set A of positive integers, we study the partition function pA(n). This function enumerates the partitions of the positive integer n into parts in A. We give simple proofs of some known and unknown identities and congruences for pA(n). For n in a special residue class, pA(n) is a polynomial in n. We examine these polynomials for linear factors, and the results are applied to a restricted m-ary partition function. We extend the domain of pA and prove a reciprocity formula with supplement. In closing we consider an asymptotic formula for pA(n) and its refinement.

scholarly journals The least common multiple of consecutive arithmetic progression terms

2011 ◽  
Vol 54 (2) ◽  
pp. 431-441 ◽  
Author(s):  
Shaofang Hong ◽  
Guoyou Qian
Keyword(s):  
Positive Integer ◽  
Periodic Function ◽  
Arithmetic Progression ◽  
Arithmetic Function ◽  
Arithmetic Progressions ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Image Position ◽  
Positive Integers

AbstractLet k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetic function gk,a,b for any positive integer n byIf we let a = 1 and b = 0, then gk,a,b becomes the arithmetic function that was previously introduced by Farhi. Farhi proved that gk,1,0 is periodic and that k! is a period. Hong and Yang improved Farhi's period k! to lcm(1, 2, … , k) and conjectured that (lcm(1, 2, … , k, k + 1))/(k + 1) divides the smallest period of gk,1,0. Recently, Farhi and Kane proved this conjecture and determined the smallest period of gk,1,0. For the general integers a ≥ 1 and b ≥ 0, it is natural to ask the following interesting question: is gk,a,b periodic? If so, what is the smallest period of gk,a,b? We first show that the arithmetic function gk,a,b is periodic. Subsequently, we provide detailed p-adic analysis of the periodic function gk,a,b. Finally, we determine the smallest period of gk,a,b. Our result extends the Farhi–Kane Theorem from the set of positive integers to general arithmetic progressions.

scholarly journals The Proof of a Conjecture Related to Divisibility Properties of z(n)

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2638
Author(s):  
Eva Trojovská ◽  
Kandasamy Venkatachalam
Keyword(s):  
Positive Integer ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Natural Density ◽  
Almost All ◽  
Divisibility Properties ◽  
Positive Integers

The order of appearance of n (in the Fibonacci sequence) z(n) is defined as the smallest positive integer k for which n divides the k—the Fibonacci number Fk. Very recently, Trojovský proved that z(n) is an even number for almost all positive integers n (in the natural density sense). Moreover, he conjectured that the same is valid for the set of integers n ≥ 1 for which the integer 4 divides z(n). In this paper, among other things, we prove that for any k ≥ 1, the number z(n) is divisible by 2k for almost all positive integers n (in particular, we confirm Trojovský’s conjecture).

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

Congruences involving alternating harmonic sums modulo pα qβ

2018 ◽  
Vol 68 (5) ◽  
pp. 975-980
Author(s):  
Zhongyan Shen ◽  
Tianxin Cai
Keyword(s):  
Positive Integer ◽  
Positive Integers ◽  
Mod P

Abstract In 2014, Wang and Cai established the following harmonic congruence for any odd prime p and positive integer r, $$\sum_{\begin{subarray}{c}i+j+k=p^{r}\\ i,j,k\in\mathcal{P}_{p}\end{subarray}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} \quad\quad(\text{mod} \,\, {p^{r}}),$$ where $ \mathcal{P}_{n} $ denote the set of positive integers which are prime to n. In this note, we obtain the congruences for distinct odd primes p, q and positive integers α, β, $$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{2pq}\end{subarray}}\frac{1}{ijk}\equiv\frac{7}{8}\left(2-% q\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}\pmod{p^{% \alpha}} $$ and $$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{pq}\end{subarray}}\frac{(-1)^{i}}{ijk}\equiv\frac{1}{2}% \left(q-2\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}% \pmod{p^{\alpha}}. $$

Sign in / Sign up

Export Citation Format

Share Document