On a divisibility problem

On a Divisibility Problem

HORSTALZER(*) - JOÂZSEFSAÂNDOR(**)

ABSTRACT- We prove that there are no integersn2andk2such thatn^kdivides W(n^k)‡sk(n). Forkˆ2this settles a conjecture of Adiga and Ramaswamy.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 11A25.

KEYWORDS. Divisibility, Euler totient, sum of divisors, Weierstrass product, inequalities.

1. Introduction

We are concerned with the classical number theoretic functions W(n) ands_a(n). Ifn1 is an integer, thenW(n) denotes the number of positive integers not execeedingnwhich are relatively prime ton. This function is known as the Euler totient. And,sa(n) denotes the sum of theath powers of the divisiors of n. Here, a is a real or complex parameter. The main properties of these and other arithmetical functions can be found, for ex- ample, in [2].

Nicol [6] and Zhang [8] were the first who studied the divisibility prob- lem

njÿ

W(n)‡s(n) : …1†

Here, as usual,sˆs₁. As each prime numbernsatisfies relation (1), this has infinitely many solutions. Let v(n) denote the number of distinct prime factors of n. If v(n)2, then the study of problem (1) is quite

(*) Indirizzo dell'A.: Morsbacher Str. 10, 51545 WaldbroÈl, Germany.

E-mail: H.Alzer@gmx.de

(**) Indirizzo dell'A.: Department of Mathematics, Babes-Bolyai University, 400084 Cluj, Romania.

E-mail: jsandor@math.ubbcluj.ro

involved. Nicol showed that the solutions of (1) are not square-free and conjectured that they are all even. He also established that if nˆ2^k3p, wherepis a prime number of the formpˆ2^kÿ27ÿ1 and k2 is an integer, thennis a solution. Zhang proved that there are no solutions of the formp^aq, wherepandqare distinct primes andais a positive integer. All casesv(n)ˆ2 andv(n)ˆ3 are settled in [5]. Also, in [5] the authors proved that for any fixed integern2 there are only finitely many odd composite solutions with v(n)ˆm, wherem2 is a fixed integer. They also obtained an asymptotic upper bound for the number of composite solutions.

Motivated by the methods and results published in [5], Harris [3], Yang [7], Jin and Tang [4] provided theorems for v(n)ˆ4 as well as related results.

In 2008, Adiga and Ramaswamy [1] investigated an analogue of prob- lem (1):

n²jÿ

W(n²)‡s2(n) : …2†

They proved that for anyn2 andv(n)3 there is no solution. Moreover, they conjectured that there is no integern2 satisfying (2).

In this note we study the following more general divisibility problem.

Letk2 be a fixed integer. Do there exist integersn2 such that n^kjÿ

W(n^k)‡sk(n) …3†

is valid? In the next section, we show that the answer to this question is ``no''.

Forkˆ2 this settles the conjecture stated by Adiga and Ramaswamy.

2. Lemmas and main Result

In order to solve the divisibility problem (3) we need three auxiliary results. The first two lemmas offer properties of Wand sk, whereas the third lemma provides an inequality involving the Weierstrass product Qⁿ

jˆ1(1ÿx_j).

LEMMA 1. Let n2and k2 be integers. If(3)is solvable, then we have

W(n^k)‡s_k(n)ˆ2n^k: …4†

PROOF. We obtain s_k(n)ˆX

djn

d^kˆX

djn

n d

_k

ˆn^kX

djn

1 d^k …5†

and X

djn

1 d^kX

dn

1 d^k5X¹

dˆ1

1

d^kˆz(k);

wherezdenotes the Riemann zeta function. Let A(n;k)ˆW(n^k)‡s_k(n) n^k :

Since W(n^k)5n^k, we get A(n;k)5z(k)‡1z(2)‡1ˆ2:64. . .. On the other hand, usings_k(n)>n^kyieldsA(n;k)>1. Thus, 15A(n;k)53. Since A(n;k) is an integer, we conclude that (4) holds. p

LEMMA2. For all integers n2and k2we have s_k(n)

n^k s2(n)

n² 5 Y

pjn;p prime

1 1ÿ1=p²: …6†

PROOF. From (5) it follows thats_k(n)=n^kis decreasing with respect to k. This leads to the first inequality in (6). Let nˆQ^r

jˆ1p_j^aj be the prime factorization ofn. Then,

s2(n) n² ˆY^r

jˆ1

p^2a_j ^j‡2ÿ1 p^2a_j ^j (p²_j ÿ1)ˆY^r

jˆ1

p²_j 1ÿ1=p^2a_j ^j‡2 p²_j ÿ1

! 5Y

pjn

p² p²ÿ1:

This settles the second inequality in (6). p

LEMMA3. Let xj2[0;1=(j‡1)] for jˆ1;. . .;r. Then we have Y^r

jˆ1

(1ÿxj)‡Y^r

jˆ1

(1ÿx²_j)^ÿ12:

The sign of equality holds if and only if x1ˆ. . .ˆxrˆ0.

PROOF. We define

F(x1;. . .;xr)ˆY^r

jˆ1

(1ÿx_j)‡Y^r

jˆ1

(1ÿx²_j)^ÿ1:

Moreover, let

Mˆ f(x₁;. . .;x_r)2R^rj0x_j1=(j‡1) (jˆ1;. . .;r)g and

(x1;...;xmaxr)2MF(x1;. . .;xr)ˆF(c1;. . .;cr):

It suffices to show that

F(c₁;. . .;c_r)2 with equality if and only ifc1ˆ. . .ˆcrˆ0.

We use induction onr. Ifrˆ1, then 0c₁1=2 and 2ÿF(c₁)ˆ c1

1ÿc²₁ 1 2

5 p ‡1

2‡c₁

! 1 2

5 p ÿ1

2ÿc₁

! 0:

The sign of equality holds if and only ifc₁ˆ0.

Next, we assume that the assertion is true for rÿ1. We define for j2 f1;. . .;rgandt2[0;1=(j‡1)]:

G(t)ˆF(c1;. . .;c_jÿ1;t;c_j‡1;. . .;cr):

Then,

0t1=(maxj‡1)G(t)ˆG(c_j):

…7†

If 05c_j51=(j‡1), then there exists a numberl2(0;1) such that c_j ˆl0‡(1ÿl) 1

j‡1: Since

G⁰⁰(t)ˆ 6t²‡2 (1ÿt²)³

Y^r

iˆ1;i6ˆj

(1ÿc²_i)^ÿ1>0;

we conclude thatGis strictly convex on [0;1=(j‡1)]. Hence, we obtain G(c_j)5lG(0)‡(1ÿl)G(1=(j‡1))

maxfG(0);G(1=(j‡1))g max

0t1=(j‡1)G(t):

This contradicts (7). Thus,

c_j2 f0;1=(j‡1)g for jˆ1;. . .;r:

We consider two cases.

CASE1. All numbersc1;. . .;crare different from 0.

Then,cjˆ1=(j‡1) (jˆ1;. . .;r) and we get F(c1;. . .;cr)ˆ 1

r‡1‡Y^r

jˆ1

1‡ 1

j(j‡2)

!

ˆ2ÿ r

(r‡1)(r‡2)52:

CASE2. At least one of the numbersc₁;. . .;cris equal to 0.

Letc_k ˆ0 withk2 f1;. . .;rg. We set

y_jˆc_j for jˆ1;. . .;kÿ1

yjˆcj‡1 for jˆk;. . .;rÿ1:

Then we have

0y_j 1

j‡1 for jˆ1;. . .;rÿ1:

Using the induction hypothesis gives

F(c1;. . .;cr)ˆF(y1;. . .;yrÿ1)2

with equality if and only ify₁ˆ. . .ˆy_rÿ1ˆ0, that is,c₁ˆ. . .ˆc_kÿ1 ˆ

c_k‡1ˆ. . .ˆc_rˆ0. p

We are now in a position to prove our main result.

THEOREM. There are no integers n2and k2satisfying relation(3).

PROOF. Using the known product representation W(n^k)=n^k ˆ Q

pjn(1ÿ1=p) as well as Lemma 2 and the prime factorization nˆQ^r

jˆ1p_j^aj we obtain

…8† W(n^k)

n^k ‡sk(n) n^k 5Y

pjn

1ÿ1 p

!

‡Y

pjn

1

1ÿ1=p²ˆY^r

jˆ1

(1ÿx_j)‡Y^r

jˆ1

(1ÿx²_j)^ÿ1

with x_jˆ1=p_j. Let p₁5p₂5 5p_r. Then, p_jj‡1 for jˆ1;. . .;r.

Applying Lemma 3 reveals that the sum on the right-hand side of (8) is less than 2. Hence,

W(n^k)‡s_k(n)52n^k:

From Lemma 1, we conclude that (3) has no solution. p Acknowledgments. We thank the referee for the careful reading of the manuscript.

REFERENCES

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Math. Anal.2(2008), pp. 1157-1161.

[2] T. M. APOSTOL, Introduction to Analytic Number Theory, Springer, New York, 1976.

[3] K. HARRIS, On the classification of integers n that divide W(n)‡s(n), J.

Number Th.129(2009), pp. 2093-2110.

[4] Q.-X. JIN - M. TANG, The 4-Nicol numbers having five different prime divisors, J. Integer Seq.14(2011), article 11.7.1 (electronic).

[5] F. LUCA- J. SAÂNDOR,On a problem of Nicol and Zhang, J. Number Th.128 (2008), pp. 1044-1059.

[6] C. A. NICOL,Some diophantine equations involving arithmetic functions, J.

Math. Anal. Appl.15(1966), pp. 154-161.

[7] S. YANG,On a divisibility problem of Nicol and Zhang, (Chinese), Adv. Math.

(China)39(2010), pp. 747-754.

[8] M. ZHANG,On a divisibility problem. J. Sichuan Univ. Nat. Sci. Ed.32(1995), pp. 240-242.

Manoscritto pervenuto in redazione il 13 Luglio 2012.