scholarly journals The Number of Unitarily k-Free Divisors of An Integer

1976 ◽  
Vol 21 (1) ◽  
pp. 19-35
Author(s):  
D. Suryanarayana ◽  
R. Sita Rama Chandra Rao
Keyword(s):  
Positive Integer ◽  
Prime Factor ◽  
Canonical Representation ◽  
Fixed Integer ◽  
Unitary Divisor ◽  
Free Divisors

Let k be a fixed integer ≧ 2. A positive integer n is called unitarily k-free, if the multiplicity of each prime factor of n is not a multiple of k; or equivalently, if n is not divisible unitarily by the k-th power of any integer > 1. By a unitary divisor, we mean as usual, a divisor d> 0 of n such that (d, n/d) = 1. The interger 1 is also considered to be unitarily k-free. The concept of a unitarily k-free integer was first introduced by Cohen (1961; §1). Let denote the set of unitarily k-free integers. When k = 2, the set coincides with the set Q* of exponentially odd integers (that is, integers in whose canonical representation each exponent is odd) discussed by Cohen himself in an earlier paper (1960; §1 and §6). A divisor d > 0 of the positive integer n is called a unitarily k-free divisor of n if d ∈ . Let (n) denote the number of unitarily k-free divisors of n.

scholarly journals Distribution of unitarily k-free integers

1975 ◽  
Vol 20 (2) ◽  
pp. 129-141 ◽  
Author(s):  
D. Suryanarayana ◽  
R. Sita Rama Chandra Rao
Keyword(s):  
Error Term ◽  
Real Variable ◽  
Canonical Representation ◽  
Order Estimate ◽  
Fixed Integer ◽  
Unitary Divisor ◽  
The Riemann Zeta Function ◽  
Elementary Methods ◽  
Riemann Zeta ◽  
Image Position

Let k be a fixed integer ≧2. A positive integer n is called unitarily k-free, if the multiplicity of each prime divisor of n is not a multiple of k; or equivalently, if n is not divisible unitarily by the kth power of any integer > 1. By a unitary divisor, we mean as usual a divisor d > 0 of n such that (d,(n/d)) = 1. The integer 1 is also considered to be unitarily k-free. These integers were first defined by Cohen (1961; § 1). Let Q*k denote the set of unitarily k-free integers. When k = 2, the set Q*2 coincides with the set Q* of exponentially odd integers (that is, integers in whose canonical representation each exponent is odd) discussed by Cohen himself in an earlier paper (1961; § 1 and § 6). Let x denote a real variable 1 and let Q*k denote the number of unitarily k-free integers ≦ x. Cohen (1961; Theorem 3.2) established by purely elementary methods that , where , the product being extended over all primes p and ζ(k) denotes the Riemann Zeta function. In the same paper Cohen (1961; Theorem 4.2) improved the order estimate of the error term in (1.1) to O(x1/k), by making use of the properties of real Dirichiet series. Later, he (Cohen; 1964) proved the same result by purely elementary methods eliminating the use of Dirichlet series.

The Fifth Unitary Perfect Number

1975 ◽  
Vol 18 (1) ◽  
pp. 115-122 ◽  
Author(s):  
Charles R. Wall
Keyword(s):  
Positive Integer ◽  
Perfect Number ◽  
Unitary Divisor ◽  
Perfect Numbers ◽  
Image Position

A divisor d of a positive integer n is a unitary divisor if d and n/d are relatively prime. An integer is said to be unitary perfect if it equals the sum of its proper unitary divisors. Subbarao and Warren [2] gave the first four unitary perfect numbers: 6, 60, 90 and 87360. In 1969,1 reported [3] thatis also unitary perfect. The purpose of this paper is to show that this last number, which for brevity we denote by W, is indeed the next unitary perfect number after 87360.

scholarly journals Square-full numbers in short intervals

1991 ◽  
Vol 110 (1) ◽  
pp. 1-3 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Positive Integer ◽  
Riemann Hypothesis ◽  
Prime Factor ◽  
Short Interval ◽  
Short Intervals ◽  
Image Position

A positive integer n is called square-full if p2|n for every prime factor p of n. Let Q(x) denote the number of square-full integers up to x. It was shown by Bateman and Grosswald [1] thatBateman and Grosswald also remarked that any improvement in the exponent would imply a ‘quasi-Riemann Hypothesis’ of the type for . Thus (1) is essentially as sharp as one can hope for at present. From (1) it follows that, for the number of square-full integers in a short interval, we havewhen and y = o (x½). (It seems more suggestive) to write the interval as (x, x + x½y]) than (x, x + y], since only intervals of length x½ or more can be of relevance here.)

scholarly journals A lower bound for a remainder term associated with the sum of digits function

1991 ◽  
Vol 34 (1) ◽  
pp. 121-142 ◽  
Author(s):  
D. M. E. Foster
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Upper Bound ◽  
Remainder Term ◽  
Fixed Integer ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
The Difference

For a fixed integer q≧2, every positive integer k = Σr≧0ar(q, k)qr where each ar(q, k)∈{0,1,2,…, q−1}. The sum of digits function α(q, k) Σr≧0ar(q, k) behaves rather erratically but on averaging has a uniform behaviour. In particular if , where n>1, then it is well known that A(q, n)∼½((q − 1)/log q)n logn as n → ∞. For odd values of q, a lower bound is now obtained for the difference 2S(q, n) = A(q, n)−½(q − 1))[log n/log q, where [log n/log q] denotes the greatest integer ≦log n /log q. This complements an upper bound already found.

scholarly journals On the Diophantine equationx3=dy2±q6

2001 ◽  
Vol 28 (8) ◽  
pp. 493-497 ◽  
Author(s):  
Fadwa S. Abu Muriefah
Keyword(s):  
Positive Integer ◽  
Prime Factor

Letq>3denote an odd prime andda positive integer without any prime factorp≡1(mod3). In this paper, we have proved that if(x,q)=1, thenx3=dy2±q6has exactly two solutions providedq≢±1(mod24).

scholarly journals On sparsely totient numbers

1991 ◽  
Vol 33 (3) ◽  
pp. 350-358
Author(s):  
Glyn Harman
Keyword(s):  
Positive Integer ◽  
Prime Factor ◽  
Multiplicative Structure ◽  
Euler Totient Function ◽  
Image Position

Following Masser and Shiu [6] we say that a positive integer n is sparsely totient ifHere φ is the familiar Euler totient function. We write ℱ for the set of sparsely totient numbers. In [6] several results are proved about the multiplicative structure of ℱ. If we write P(n) for the largest prime factor of n then it was shown (Theorem 2 of [6]) thatand 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).

scholarly journals Bi-unitary multiperfect numbers, IV(a)

2020 ◽  
Vol 26 (4) ◽  
pp. 2-32
Author(s):  
Pentti Haukkanen ◽  
◽  
Varanasi Sitaramaiah
Keyword(s):  
Positive Integer ◽  
Unitary Divisor

A divisor d of a positive integer n is called a unitary divisor if \gcd(d, n/d)=1; and d is called a bi-unitary divisor of n if the greatest common unitary divisor of d and n/d is unity. The concept of a bi-unitary divisor is due to D. Surynarayana (1972). Let \sigma^{**}(n) denote the sum of the bi-unitary divisors of n. A positive integer n is called a bi-unitary multiperfect number if \sigma^{**}(n)=kn for some k \geq 3. For k=3 we obtain the bi-unitary triperfect numbers. Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers. The present paper is Part IV(a) in a series of papers on even bi-unitary multiperfect numbers. In parts I, II and III we found all bi-unitary triperfect numbers of the form n=2^{a}u, where 1\leq a \leq 6 and u is odd. There exist exactly ten such numbers. In this part we solve partly the case a=7. We prove that if n is a bi-unitary triperfect number of the form n=2^{7}.5^{b}.17^{c}.v, where (v, 2.5.17)=1, then b\geq 2. We then confine ourselves to the case b=2. We prove that in this case we have c=1 and further show that n=2^{7}.3^{2}.5^{2}.7.13.17=44553600 is the only bi-unitary triperfect number of this form.

The Fractal Dimension of Sets Derived from Complex Bases

1986 ◽  
Vol 29 (4) ◽  
pp. 495-500 ◽  
Author(s):  
William J. Gilbert
Keyword(s):  
Fractal Dimension ◽  
Positive Integer ◽  
Positive Root ◽  
Complex Numbers ◽  
Integer Part ◽  
Fixed Integer

AbstractFor each positive integer n, the radix representation of the complex numbers in the base —n + i gives rise to a tiling of the plane. Each tile consists of all the complex numbers representable in the base -n + i with a fixed integer part. We show that the fractal dimension of the boundary of each tile is 2 log λn/log(n2 + 1), where λn is the positive root of λ3 - (2n - 1) λ2 - (n - 1) 2λ - (n2 + 1).

scholarly journals Periodic rings with commuting nilpotents

1984 ◽  
Vol 7 (2) ◽  
pp. 403-406
Author(s):  
Hazar Abu-Khuzam ◽  
Adil Yaqub
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Finite Fields ◽  
Boolean Ring ◽  
Fixed Integer ◽  
Integer Coefficients ◽  
Nilpotent Elements ◽  
Periodic Rings

LetRbe a ring (not necessarily with identity) and letNdenote the set of nilpotent elements ofR. Suppose that (i)Nis commutative, (ii) for everyxinR, there exists a positive integerk=k(x)and a polynomialf(λ)=fx(λ)with integer coefficients such thatxk=xk+1f(x), (iii) the setIn={x|xn=x}wherenis a fixed integer,n>1, is an ideal inR. ThenRis a subdirect sum of finite fields of at mostnelements and a nil commutative ring. This theorem, generalizes the “xn=x” theorem of Jacobson, and (takingn=2) also yields the well known structure of a Boolean ring. An Example is given which shows that this theorem need not be true if we merely assume thatInis a subring ofR.

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