Some arithmetical properties of m-ary partitions

1970 ◽  
Vol 68 (2) ◽  
pp. 447-453 ◽  
Author(s):  
Öystein Rödseth
Keyword(s):  
Partition Function ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Fixed Integer ◽  
De Bruijn ◽  
Image Position

We denote by tm(n) the number of partitions of the positive integer n into non-decreasing parts which are positive or zero powers of a fixed integer m > 1 and we call tm(n) ‘the m-ary partition function’. Mahler(1) obtained an asymptotic formula for tm(n), the first term of which isMahler's result was later improved by de Bruijn (2).

Congruence properties of the binary partition function

1969 ◽  
Vol 66 (2) ◽  
pp. 371-376 ◽  
Author(s):  
R. F. Churchhouse
Keyword(s):  
Partition Function ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Functional Equations ◽  
Precise Calculation ◽  
Binary Partition ◽  
Powers Of 2 ◽  
De Bruijn ◽  
Congruence Properties

We denote by b(n) the number of ways of expressing the positive integer n as the sum of powers of 2 and we call b(n) ‘the binary partition function’. This function has been studied by Euler (1), Tanturri (2–4), Mahler (5), de Bruijn(6) and Pennington (7). Euler and Tanturri were primarily concerned with deriving formulae for the precise calculation of b(n), whereas Mahler deduced an asymptotic formula for log b(n) from his analysis of the functions satisfying a certain class of functional equations. De Bruijn and Pennington extended Mahler's work and obtained more precise results.

On the kth root partition function

2021 ◽  
pp. 1-15
Author(s):  
Ya-Li Li ◽  
Jie Wu
Keyword(s):  
Partition Function ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Real Number ◽  
Number Of Solutions ◽  
Asymptotic Development ◽  
Number Formula

For any positive integer [Formula: see text], let [Formula: see text] be the number of solutions of the equation [Formula: see text] with integers [Formula: see text], where [Formula: see text] is the integral part of real number [Formula: see text]. Recently, Luca and Ralaivaosaona gave an asymptotic formula for [Formula: see text]. In this paper, we give an asymptotic development of [Formula: see text] for all [Formula: see text]. Moreover, we prove that the number of such partitions is even (respectively, odd) infinitely often.

On the decimal representation of integers

1952 ◽  
Vol 48 (4) ◽  
pp. 555-565 ◽  
Author(s):  
M. P. Drazin ◽  
J. Stanley Griffith
Keyword(s):  
Positive Integer ◽  
Fixed Integer ◽  
Usual Convention ◽  
Representation Of Integers ◽  
Image Position

Let r be any fixed integer with, r≥ 2; then, given any positive integer n, we can find* integers αk(r, n) (k = 0, 1, 2, …) such thatwhere, subject to the conditionsthe integers αk(r, n) are uniquely determined, and, in fact, clearlyαk(r, n) = [n/rk] − r[n/rk+1](square brackets denoting integral parts, according to the usual convention).

The Circle Problem in an Arithmetic Progression

1968 ◽  
Vol 11 (2) ◽  
pp. 175-184 ◽  
Author(s):  
R.A. Smith
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Arithmetic Progression ◽  
Error Term ◽  
Circle Problem ◽  
Image Position

In following a suggestion of S. Chowla to apply a method of C. Hooley [3] to obtain an asymptotic formula for the sum ∑ r(n)r(n+a), where r(n) denotes the number of representations of n≤xn as the sum of two squares and is positive integer, we have had to obtain non-trivial estimates for the error term in the asymptotic expansion of1

scholarly journals On the Distribution of r-Tuples of k-Free Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Ting Zhang ◽  
Huaning Liu
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Fixed Integer

For k, being a fixed integer ≥2, a positive integer n is called k-free number if n is not divisible by the kth power of any integer >1. In this paper, we studied the distribution of r-tuples of k-free numbers and derived an asymptotic formula.

XVII.—On the Asymptotic Expansion of the Characteristic Numbers of the Mathieu Equation

1930 ◽  
Vol 49 ◽  
pp. 210-223 ◽  
Author(s):  
Sydney Goldstein
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Mathieu Equation ◽  
Characteristic Number ◽  
Numerical Approximations ◽  
Characteristic Numbers ◽  
The Difference ◽  
Image Position ◽  
Integral Order

An asymptotic formula has recently been given for the characteristic numbers of the Mathieu equation From tabular values, it will be seen that the formula provides good numerical approximations to the characteristic numbers of integral order; but as pointed out by Ince, it provides better approximations to the characteristic numbers of order (m + ½), where m is a positive integer or zero. In this paper we shall first attempt to find out why this should be so, and then go on to show that the formula is probably an asymptotic expansion, in the Poincaré sense, for any characteristic number. A new asymptotic formula is then found for the difference between two characteristic numbers.

scholarly journals Estimates for a remainder term associated with the sum of digits function

1987 ◽  
Vol 29 (1) ◽  
pp. 109-129 ◽  
Author(s):  
D. M. E. Foster
Keyword(s):  
Positive Integer ◽  
Remainder Term ◽  
Fixed Integer ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
Image Position ◽  
Average Behaviour

If q(≥2) is a fixed integer it is well known that every positive integer k may be expressed uniquely in the formWe introduce the ‘sum of digits’ functionBoth the above sums are of course finite. Although the behaviour of α(q, k) is somewhat erratic, its average behaviour is more regular and has been widely studied.

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 TERMS IN A QUADRATIC PROGRESSION

2012 ◽  
Vol 86 (3) ◽  
pp. 389-404 ◽  
Author(s):  
GUOYOU QIAN ◽  
QIANRONG TAN ◽  
SHAOFANG HONG
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Periodic Function ◽  
Arithmetic Function ◽  
Local Analysis ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Image Position

AbstractLet k be any given positive integer. We define the arithmetic function gk for any positive integer n by We first show that gk is periodic. Subsequently, we provide a detailed local analysis of the periodic function gk, and determine its smallest period. We also obtain an asymptotic formula for log lcm0≤i≤k {(n+i)2+1}.

scholarly journals GENERALIZED D. H. LEHMER PROBLEM OVER SHORT INTERVALS

2010 ◽  
Vol 53 (2) ◽  
pp. 293-299 ◽  
Author(s):  
PING XI ◽  
YUAN YI
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Asymptotic Properties ◽  
Gauss Sums ◽  
Kloosterman Sums ◽  
Short Intervals ◽  
Lehmer Problem ◽  
Consecutive Integers ◽  
Image Position ◽  
Fixed Positive Integer

AbstractLet n ≥ 2 be a fixed positive integer, q ≥ 3 and c, ℓ be integers with (nc, q)=1 and ℓ|n. Suppose and consist of consecutive integers which are coprime to q. We define the cardinality of a set: The main purpose of this paper is to use the estimates of Gauss sums and Kloosterman sums to study the asymptotic properties of N(, , c, n, ℓ; q), and to give an interesting asymptotic formula for it.

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