1Introduction Aboutk-perfectnumbers

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About k-perfect numbers

Mih´aly Bencze

Abstract

ABSTRACT. In this paper we present some results aboutk-perfect numbers, and generalize two inequalities due to M. Perisastri (see [6]).

1 Introduction

Definition. A positive integer n is k-perfect if σ(n) = kn, when k > 1, k ∈ Q. The special case k = 2 corresponds to perfect numbers, which are intimately connected with Mersenne primes. We have the following smallest k-perfect numbers. For k = 2 (6,28,496,8128, ...), for k = 3 (120,672,523776,459818240, ...),fork= 4 (30240,32760,2178540, ...),fork= 5 (14182439040,31998395520, ...),fork= 6 (154345556085770649600, ...).

For a given prime numberp, ifnisp-perfect andpdoes not dividen, then pn id (p+ 1)−perfect. This imples that an integernis a 3−perfect number divisible by 2 but not by 4, if and only if ⁿ₂ is an odd perfect number, of which none are known. If 3nis 4k−perfect and 3 does not dividen, thenn is 3k−perfect.

Ak−perfect number is a positive integernsuch that its harmonic sum of divisors isk.

For the perfect numbers we have the followings: 28 = 1³ + 3³, 496 = 1³+ 3³+ 5³+ 7³,8128 = 1³+ 3³+ 5³+ 7³+ 9³+ 11³+ 13³+ 15³ etc. We posted the following conjecture:

Conjecture. (Bencze, M., 1978) Ifnisk-perfect, then exist odd positive integers u_i (i= 1,2, ..., r) such that

Key Words: perfect numbers

2010 Mathematics Subject Classification: 11A25 Received: May, 2013.

Revised: September, 2013.

Accepted: November, 2013.

45

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n=

r

X

i=1

u^k+1_i

MAIN RESULTS

Theorem 1. If f : R → R is convex and increasing, N =p^α₁¹p^α₂²...p^α_nⁿ written in cannonical form isk-perfect, then:

n

X

i=1

f 1

p_i

≥





 nf

n

q3 2−1

if N is even nf

3n√ k²−1

if N is odd Proof. IfN is even then it follows

n

Y

i=1

pi+ 1 p_i > 3

2 Forx≥3 holds^x+1_x ≥ ³

r x

x−1

²

(see [9]), therefore ifNis odd then yields

n

Y

i=1

pi+ 1 p_i > ³

v u u t

n

Y

i=1

pi

p_i−1 ²

> ³

√ k²

because

n

Y

i=1

pi

pi−1 =k

n

Y

i=1

p^α_iⁱ⁺¹ p^α_iⁱ⁺¹−1 > k Using the AM-GM inequality we obtain:

n

Y

i=1

pi+ 1

p_i ≤ 1

n

X

i=1

pi+ 1 p_i

!n

= 1 + 1 n

n

X

i=1

1 p_i

!n

Finally

n

X

i=1

1 pi

>





 n

n

q3 2−1

if N is even n_3n√

k²−1

if N is odd

Because f is convex and increasing from Jensen’s inequality we get

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n

X

i=1

f 1

pi

≥nf 1 n

n

X

i=1

1 pi

!

≥





 nf

n

q3 2−1

if N is even nf_3n√

k²−1

if N is odd (1)

Theorem 2. If g : R →R is convex and increasing, N = p^α₁¹p^α₂²...p^α_nⁿ written in cannonical form isk-perfect, then:

n

X

i=1

g 1

p_i

≤





 ng

1−qⁿ

6 kπ²

if N is even ng

1−qⁿ

8 kπ²

if N is odd Proof. We have the following:

n

Y

i=1

pi

p_i−1 =

n

Y

i=1

p^α_iⁱ⁺¹−1 (p_i−1)p^α_iⁱ

n

Y

i=1

p^α_iⁱ⁺¹ p^α_iⁱ⁺¹−1 =k

n

Y

i=1

1 1− ¹

p^αi_i ⁺¹

=k

n

Y

i=1





∞

X

j=0

1 p_i

^j



≤

≤k

n

Y

i=1





∞

X

j=0

1 p^2j_i



<





 k

∞

P

n=1 1

n² if N is even k

∞

P

n=0 1

(2n+1)² if N is odd

= ( kπ²

6 if N is even

kπ²

8 if N is odd From AM-GM inequality yields

n

Y

i=1

p_i pi−1 ≥





 n

n

P

i=1 p_i−1

p_i







n

=





 n n−

n

P

i=1 1 p_i







n

therefore

n

X

i=1

1 pi

<





 n

1− qⁿ

6 kπ²

if N is even n

1− qⁿ

8 kπ²

if N is odd According to Jensen’s inequality yields

n

X

i=1

g 1

pi

≤ng 1 n

n

X

i=1

1 pi

!

≤





 ng

1−qⁿ

6 kπ²

if N is even ng

1−qⁿ

8 kπ²

if N is odd

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Corolloary 1. IfN =p₁ p₂ ...p_nⁿwritten in cannonical form isk-perfect then:





 n

n

q3 2−1

if N is even n_3n√

k²−1

if N is odd

<

n

X

i=1

1 pi

<





 n

1− qⁿ

6 kπ²

if N is even n

1− qⁿ

8 kπ²

if N is odd Theorem 3. Ifx, t >0 then

(x+ 1)t^x+1¹ −xt¹^x ≤1

Proof. Fort= 1 we have the equality. Let 0 < t <1. Since the function u(x) = xt¹^x is continuous and differentiable we can apply the Lagrange’s theorem and we obtain

(x+ 1)t^x+1¹ −xt^x¹

(x+ 1)−x =u(x+ 1)−u(x)

(x+ 1)−x =u⁰(z) whenx < z < x+ 1 hence we have the inequality

t^z¹

1−1 zlnt

<1 or 1−1

zlnt < t⁻¹^z. Developingt⁻¹^z into McLauren’s series it results

1−1

zlnt <1− 1

1!zlnt+ 1

2!z²ln²t− 1

3!z³ln³t+...

or

∞

X

r=2

(−1)^rln^rt r!z^r >0 or

∞

X

r=2

ln^r¹_t r!z^t >0

that is obvious because ln¹_t > 0 due to ¹_t > 1. Let be t > 1. Then is enough to show that the functionV (x) =x

t^x¹ −1

is decreasing.

DifferentiableV we get V⁰(x) =t^x¹ −t¹^x · 1

xlnt−1 =−

∞

X

r=2

ln^rt x^r(r−1)!

1−1

r

<0

Since V is decreasing and we may say that V(x+ 1) < V (x) hence and from it follows the inequality of the ennunciation.

Corollary 2. IfN =p^α₁¹p^α₂²...p^α_nⁿ is a k-perfect number written in cannonical form, then:

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ln³₂ if N is even

2

3lnk if N is odd <

n

X

i=1

1 pi

<

( ln^kπ₆² if N is even ln^kπ₈² if N is odd Proof. Using the Theorem 3 it is proved that the series

n ⁿ

r3 2 −1

!!

n∈N^∗

and n_3n√

k²−1

n∈N^∗

are decreasing, and the series n 1− ⁿ

r 6 kπ²

!!

n∈N^∗

and n 1− ⁿ r 8

kπ²

!!

n∈N^∗

are increasing. It means that the minimum and maximum are reached only thenn→ ∞.

Sincen → ∞we have 0· ∞. That is why L’Hospital rule and so we find the results of the enunciation.

Remark 1. For k= 2 we reobtain the M.Perisastri’s inequality

n

X

i=1

1 pi

<2 lnπ 2 (see [6]).

Corollary 3. Let N = p^α₁¹p^α₂²...p^α_nⁿ be a k−perfect number written in cannonical form and Pmax = {p1, p2, ..., pn} and Pmin = min{p1, p2, ..., pn}, then

Pmin<

( 1

n√

3

2−1 if N is even

1

3n√

k²−1 if N is odd and

P_max>







1 1−ⁿq

6 kπ2

if N is even

1 1−ⁿq

8 kπ2

if N is odd Proof. Considering that

n Pmax

<

n

X

i=1

1 pi

respective

n

X

i=1

1 pi

< n Pmin

from the theorem if follows the affirmation.

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Remark 2. Let N =p₁ p₂ ...p_nⁿ be a k-perfect number written in cannonical form, then

Pmin< 2n k²−1 + 2 (see the method of M. Perisastri’s)

Acknowledgements. The author wishes to express his gratitude to the Or- ganizing Committee of the workshop ”Workshop on Algebraic and Analytic Number Theory and their Applications” (PN-II-ID-WE - 2012 - 4 -161). The publication of this paper is supported by the grant of CNCS-UEFISCDI (Ro- manian National Authority for Scientific Research): PN-II-ID-WE - 2012 - 4 -161.

References

[1] B. Apostol, Extremal orders of some functions connected to regular integers modulon,An. St. Univ. Ovidius Constanta, Vol. 21(2),2013, 5-19.

[2] M. Bencze, On perfect numbers, Studia Mathematica, Univ. Babes- Bolyai, Nr. 4, 1981, 14-18.

[3] G. Hardy, D.E. Littlewood, G. Polya,Inequalities, Cambridge, University Press, 1964.

[4] H.J. Kanold, Uber mehrfach volkommene Zahlen, II. J. Reine Angew.¨ Math., 1957, 197, 82-96.

[5] Octogon Mathematical Magazine (1993-2013).

[6] M. Perisastri,A note on odd perfect numbers, The Mathematics Student 26(1958), 179-181.

[7] J. S´andor, B. Cristici, eds.: Handbook of number theory II, Dordrecht, Kluwer Academic, 2004.

[8] W. Sierpinski,Elementary theory of numbers,Warsawa, 1964.

[9] The American Mathematical Monthly, E.2308(1971), E.2162(1969) (Simeon Reich’s note).

Mih´aly Bencze, Str. H˘armanului 6,,

505600 S˘acele-N´egyfalu, Jud. Bra¸sov, Romania Email: [email protected]