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.

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.

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.

ON CONJUGATES OF COLLATZ-TYPE MAPPINGS

2008 ◽  
Vol 04 (01) ◽  
pp. 117-120
Author(s):  
STEFAN KOHL
Keyword(s):  
Positive Integer ◽  
Residue Class ◽  
Affine Mapping ◽  
Absolute Value ◽  
Residue Classes ◽  
Almost Everywhere ◽  
Finite Set ◽  
The Absolute ◽  
Almost All ◽  
Original Motivation

A mapping f : ℤ → ℤ is called residue-class-wise affine if there is a positive integer m such that it is affine on residue classes (mod m). If there is a finite set S ⊂ ℤ which intersects nontrivially with any trajectory of f, then f is called almost contracting. Assume that f is a surjective but not injective residue-class-wise affine mapping, and that the preimage of any integer under f is finite. Then f is almost contracting if and only if there is a permutation σ of ℤ such that fσ = σ-1 ◦ f ◦ σ is either monotonically increasing or monotonically decreasing almost everywhere. In this paper it is shown that if there is no positive integer k such that applying f(k) decreases the absolute value of almost all integers, then σ cannot be residue-class-wise affine itself. The original motivation for the investigations in this paper comes from the famous 3n + 1 Conjecture.

Asymptotic formulae in the theory of partitions

1954 ◽  
Vol 50 (2) ◽  
pp. 225-241 ◽  
Author(s):  
C. B. Haselgrove ◽  
H. N. V. Temperley
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Classical Method ◽  
Contour Integration ◽  
Large Positive Integer ◽  
Asymptotic Formulae ◽  
Positive Integers

It is the object of this paper to obtain an asymptotic formula for the number of partitions pm(n) of a large positive integer n into m parts λr, where the number m becomes large with n and the numbers λ1, λ2,… form a sequence of positive integers. The formula is proved by using the classical method of contour integration due to Hardy, Ramanujan and Littlewood. It will be necessary to assume certain conditions on the sequence λr, but these conditions are satisfied in most of the cases of interest. In particular, we shall be able to prove the asymptotic formula in the cases of partitions into positive integers, primes and kth powers for any positive integer k.

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 PARITY OF PARTITION FUNCTIONS

2003 ◽  
Vol 14 (04) ◽  
pp. 437-459 ◽  
Author(s):  
BRUCE C. BERNDT ◽  
AE JA YEE ◽  
ALEXANDRU ZAHARESCU
Keyword(s):  
Partition Function ◽  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Lower Bounds ◽  
Partition Functions ◽  
New Methods ◽  
Formal Power ◽  
Positive Integers

Let S denote a subset of the positive integers, and let pS(n) be the associated partition function, that is, pS(n) denotes the number of partitions of the positive integer n into parts taken from S. Thus, if S is the set of positive integers, then pS(n) is the ordinary partition function p(n). In this paper, working in the ring of formal power series in one variable over the field of two elements Z/2Z, we develop new methods for deriving lower bounds for both the number of even values and the number of odd values taken by pS(n), for n ≤ N. New very general theorems are obtained, and applications are made to several partition functions.

scholarly journals Collatz Dynamics is Partitioned by Residue Class Regularly

2020 ◽  
Author(s):  
Wei Ren
Keyword(s):  
Positive Integer ◽  
Residue Class ◽  
Collatz Conjecture ◽  
If Current ◽  
Positive Integers

<div>We propose Reduced Collatz Conjecture that is equivalent to Collatz</div><div>Conjecture, which states that every positive integer can return to</div><div>an integer less than it, instead of 1. Reduced Collatz Conjecture</div><div>should be easier because some properties are presented in reduced</div><div>dynamics, rather than in original dynamics (e.g., ratio and period).</div><div>Reduced dynamics is a computation sequence from starting integer to</div><div>the first integer less than it, and original dynamics is a</div><div>computation sequence from starting integer to 1. Reduced dynamics is</div><div>a component of original dynamics. We denote dynamics of x as a</div><div>sequence of either computations in terms of ``I'' that represents</div><div>(3*x+1)/2 and ``O'' that represents x/2. Here 3*x+1 and x/2 are</div><div>combined together, because 3*x+1 is always even and followed by x/2.</div><div>We formally prove that all positive integers are partitioned into</div><div>two halves and either presents ``I'' or ``O'' in next ongoing</div><div>computation. More specifically, (1) if any positive integer x that</div><div>is i module $2^t$ (i is an odd integer) is given, then the first t</div><div>computations (each one is either ``I'' or ``O'' corresponding to</div><div>whether current integer is odd or even) will be identical with that</div><div>of i. (2) If current integer after t computations (in terms of ``I''</div><div>or ``O'') is less than x, then reduced dynamics of x is available.</div><div>Otherwise, the residue class of x (namely, i module $2^t$) can be</div><div>partitioned into two halves (namely, i module $2^{t+1}$ and $i+2^t$</div><div>module $2^{t+1}$), and either half presents ``I'' or ``O'' in</div><div>intermediately forthcoming (t+1)-th computation.</div>

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.

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).

Sign in / Sign up

Export Citation Format

Share Document