Multiplicatively Perfect and Related Numbers

Journal of Rajasthan Academy of Physical Sciences ISSN : 0972-6306; URL : http://raops.org.in Vol.15, No.4, December, 2016, 345-350

MULTIPLICATIVELY PERFECT AND RELATED NUMBERS

Shikha Yadav* and Surendra Yadav** *Department of Mathematics and Astronomy, University of Lucknow, Lucknow- 226007, U.P., India E-mail: [email protected] **Department of Applied Mathematics, Babasaheb Bhimrao Ambedkar University, Lucknow-226025, U.P., India, E-mail: ssyp_ [email protected]

Abstract: Sandor [4] has discussed multiplicatively perfect (푀perfect) and multiplicatively +perfect numbers (called as 푀 +perfect). In this paper, we have discussed a new perfect number called as multiplicatively – perfect numbers (also called as 푀 − perfect). Further we study about Abundant, Deficient, Harmonic and Unitary analogue of harmonic numbers. Keywords and Phrases: Multiplicatively perfect numbers, Abundant numbers, Deficient numbers, Harmonic numbers and unitary divisor. 2010 Mathematics Subject Classification: 11A25.

1. Introduction Sandor and Egri [3], Sandor [4] have defined multiplicatively perfect and multiplicatively + perfect numbers as follows, A positive integer 푛 ≥ 1 is called multiplicatively perfect if 푅 푛 = 푛2, (1) where, 푅 푛 is product of divisor function. Let 푑1, 푑2, 푑3, … 푑푟 are the divisors of 푛 then

푅 푛 = 푑1. 푑2. 푑3. … 푑푟 . 푑(푛 ) Also, 푅 푛 = 푛 2 (2)

Let 푅+(푛) denotes the product of even divisors of 푛. We say that 푛 is 푀 + perfect number if 2 푅+ 푛 = 푛 . (3) Now, in this paper we define 푀 − perfect numbers as follows, let 푅− 푛 denotes the product of odd divisors of 푛 .We say that 푛 is 푀 − perfect if

346 Shikha Yadav and Surendra Yadav

2 푅− 푛 = 푛 . (4) Sandor and Egri[3] have discussed all forms of 푀perfect and 푀 +perfect numbers by following theorem. 3 Theorem 1.1 All 푀perfect numbers are of the form 푛 = 푝1푝2 or 푛 = 푝1 , where 푝1 ≠ 푝2 are arbitrary primes. 3 Theorem 1.2 All 푀 +perfect numbers are of the form 푛 = 8푝1푝2 or 푛 = 8푝1 , where 푝1 ≠ 푝2 are distinct odd primes. In this paper we discuss the form of 푀 − perfect numbers in section 2. Sandor and Egri [3] called 푛 to be nobly abundant if both 휎 푛 and 푑(푛) are abundant numbers. Similarly 푛 is called nobly deficient if both 휎 푛 and 푑(푛) are deficient, 푛 is called abundant if 휎 푛 > 2푛 and deficient if 휎 푛 < 2푛. Sandor and Egri [3] introduced a number 푛 ∈ 푁 is called harmonic number if 휎 푛 /푛푑(푛). Many results on harmonic numbers have been proved in Sandor and Egri[3] introduced by Cohen and Sorli [2] or Cohen and Deng [1].

Let 휎푒 (푛) and 푑푒 (푛) denotes the sum and product of all 푒 - divisors of 푛. Also that 휎푒 1 = 1, 푑푒 1 = 1 . Straus and Subbarao [5] called 푛 to be 푒- perfect if

휎푒 푛 = 2푛 . (5) Further, Sandor and Egri [3] introduced that 푛 is called 푒 -Harmonic of type 1, if

휎푒 (푛)/푛푑푒 푛 , (6) and 푒 -Harmonic of type 2, if 푝푒 (푛)/푛푑푒 (푛) where 푟 푎푖−푑푖 푎1 푎2 푎푟 푝푒 푛 = 푖=1( 푑푖/푎푖 푝푖 ) , 푛 = 푝1 . 푝2 … . . 푝푟 (7) 푎−푑 푎 And 푝푒 푛 = 푑/푎 푝 , 푛 = 푝 . (8) Sandor defined that 푛 ∈ 푁 is Modified 푒 -perfect number if

푝푒 (푛)/푛 . (9) Sandor and Egri [3] defined Geometric numbers as follows, let 퐺(푛) is geometric mean of divisors of an integer 푛 given by

1 퐺 푛 = [푅 푛 ]푑 푛 . (10) A positive integer 푛 is called geometric number if 퐺 푛 is an integer.It is clear using (1.1) that 퐺 푛 = 푛 , (11) so all Geometric numbers are perfect squares.

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In Mladen, Vassilev Missana and Krassimir Atanassov [6] definition of 푒-perfect number is given as follows:

A positive integer 푛 is called 푒 -perfect number if 휎푒 푛 = 2푛 , (12) where 휎푒 푛 is the sum of all 푒 divisors of 푛. In Cohen and Sorli [2] many results on harmonic numbers has been given, if we consider unitary divisors of 푛 then unitary harmonic numbers can be defined as, A number 푛 ≥ 1 is called unitary harmonic if 휎∗(푛)/푛푑∗(푛) . (13) where 푑∗(푛) is the number of unitary divisors of 푛 which is defined as 푑∗ 푛 = 2푟 , (14)

훼1 훼2 훼푟 ∗ if 푛 = 푝1 . 푝2 … . . 푝푟 , and 휎 (푛) is unitary divisor function. Now, 푅∗(푛) is the product of unitary divisors of 푛 which is defined as,

푑∗ (푛) 푅∗ 푛 = (푛) 2 . (15) Now, unitary geometric mean of a positive integer is defined as,

1 퐺∗ 푛 = (푅∗ 푛 )푑∗ (푛), (16) and a positive integer 푛 is called unitary geometric number if 퐺∗ 푛 is an integer so using (15), we have 퐺∗ 푛 = 푛 . (17) Hence a unitary geometric number is infact a perfect square. In section 2 we prove some results about the related numbers discussed as above in this section. 2. Results 3 Theorem 2.1 All 푀 − perfect numbers are of the form 푛 = 푝1푝2 or 푛 = 푝1 , where 푝1 ≠ 푝2 are distinct odd primes. Proof: Let 푛 = 2푘 푁 is a 푀 − perfect number, where 푁 is odd positive integer. Then all odd divisors 푑 of 푛 are 푑 = 푁1,where 푁1/푁 .

Therefore, 푅− 푛 = 푁1/푁 푁1 = 푅 푁 . Using (1.4) we get 푅 푁 = 22푘 푁2 , because 푅 푁 is always odd as 푁 is odd therefore 푘 = 0, so 푛 = 푁, this implies 푅 푁 = 푁2, thus 푁 is odd 푀 perfect number. 3 According to theorem (1.1), 푛 = 푝1푝2 or 푛 = 푝1 . Theorem 2.2 Let 푟 ≥ 3 and 푟 = 2푘 − 1 be a mersenne prime then 푛 = 2푟 has the property that 푑(푛) is deficient , 휎(푛) is abundant and if 푟 ≥ 7 then 푛 is deficient. If 푟 < 7 then 푛 is a perfect number.

348 Shikha Yadav and Surendra Yadav

Proof: Let 푛 = 2푟 , where 푟 ≥ 3 and 푟 = 2푘 − 1 be a mersenne prime. Then, 휎 푑 푛 = 휎 4 = 7 < 2. 푑 푛 = 8, so, 휎 푑 푛 < 2푑 푛 . Hence 푑 푛 is deficient. Now, 휎 휎 푛 = 휎 휎 2푟 = 휎 휎 푟 휎(2) = 휎 3. 2푘 = 휎 3 휎 2푘 = 4(2푘+1 − 1). Since, 2 휎 2푟 =2휎 2 휎(푟)= 3. 2푘+1 < 4 2푘+1 − 1 , ∀ 푘 ≥ 2. Therefore, 휎 휎 푛 > 2휎 푛 . Hence, 휎(푛) is abundant. Since, 휎 푛 = 휎 2푟 = 3. 2푘 < 4 2푘 − 1 = 2푛, ∀ 푘 > 2. Therefore, 휎 푛 < 2푛 . Hence, 푛 is deficient. Since, 푟 = 3 and 푛 = 6 for 푘 = 2 therefore 푛 is a perfect number. Hence, for 푟 < 7, 푛 is a perfect number. Theorem 2.3 Let 푟 ≥ 3 and 푞 = 푟2 + 푟 + 1 is a prime then 푛 = 2푟2 has property that 휎(푛) is deficient, 푑(푛) is perfect and 푛 is abundant. Proof: Let 푛 = 2푟2 , 푟 ≥ 3 is a prime number. Then, 휎 푛 = 휎 2푟2 = 휎 2 휎 푟2 = 3 푟2 + 푟 + 1 = 3푞, where, 푞 = 푟2 + 푟 + 1 be a prime. Now, 휎 휎 푛 = 휎 3. 푞 = 휎 3 휎 푞 = 4 푞 + 1 = 4 푟2 + 푟 + 2 , and 2휎 푛 = 2휎 2푟2 = 2.3 푟2 + 푟 + 1 = 6 푟2 + 푟 + 1 . Since, it is clear that 4 푟2 + 푟 + 2 < 6 푟2 + 푟 + 1 . For 푟 ≥ 3. Therefore, 휎 휎 푛 < 2휎 푛 . Hence 휎 푛 is deficient. Since, 휎 푑 푛 = 휎 푑 2푟2 = 휎 푑 2 푑 푟2 = 휎 2.3 = 휎 6 = 12 = 2. 푑(2푟2)휎 푑 푛 = 2푑 푛 . Hence 푑 푛 is perfect. Since, 휎 2푟2 = 휎 2 휎 푟2 = 3 푟2 + 푟 + 1 > 4푟2, for 푟 ≥ 3. Therefore, 휎 푛 > 2푛 . Hence, 푛 is abundant. Preposition 2.1 If 푛 = 8 then 휎(푛) and 푑(푛) both are deficient i.e. 푛 = 8 is a nobly deficient number. Proof: Since, 휎 휎 8 = 휎 15 = 24 < 2휎 8 = 2.15 = 30, therefore, 휎 8 is deficient. Since, 휎 푑 8 = 휎 4 = 7 < 2푑 8 = 2.4 = 8, therefore, 푑 8 is deficient. Hence 푛 = 8 is nobly deficient.

Theorem 2.5 All multiplicatively Perfect numbers of the form 푛 = 푝1푝2 are modified 푒-Perfect numbers.

Proof: Using (8), since, 푝푒 푝1푝2 = 푝푒 푝1 푝푒 푝2 = 1, therefore, (9) holds true. Hence 푛 is a Modified 푒-Perfect number.

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Theorem 2.6 All multiplicatively Perfect numbers of the form 푛 = 푝1푝2 are 푒-Harmonic of both types. Proof: If 푛 is product of distinct primes then 푛 is 푒-Harmonic of both types (proof has been given in Sandor and E. Egri [8]). Hence 푛 = 푝1푝2 is 푒-Harmonic of both types. Theorem 2.7 If 푝 is an arbitrary prime, then 푛 = 푝 is 푒-Harmonic of both types and modified 푒-Perfect.

Proof: For, 푛 = 푝 , 푝푒 푝 = 1. Hence (9) holds true. Also by the proof given in Sandor and Egri [3], 푛 = 푝 is 푒-Harmonic of both types. Theorem 2.8 If 푛 = 푝훼 , where 훼 is an even positive integer.Then 푛 is a geometric number. Proof: Let 푛 = 푝훼 , where 훼 is an even positive integer. Let 훼 = 2푟, then 푛 = 푝2푟 using (1.1), we get

2푟+1 2푟+1 1 1 푅 푛 = (푝2푟 ) 2 , now using (1.10), we get 퐺 푛 = [(푝2푟 ) 2 ] 2푟+1 = (푝2푟 )2 = 푝푟 = 푝2푟 , which is always an integer. Hence 푛 = 푝훼 , where 훼 is an even positive integer and is a geometric number. Theorem 2.9 All prime numbers are modified 푒-Perfect numbers but not a geometric number.

Proof: Let 푛 = 푝 , where 푝 is a prime number. Since 푝푒 푝 = 1, therefore, using (9) 푛 is a modified 푒-Perfect number. Since 푅 푝 = 푝, therefore, using (10), 퐺 푛 is not an integer. Hence 푛 = 푝 is not a geometric number. 2 2 4 2 Remark- Since, 푝푒 36 = 12 and 푝푒 144 = 36 therefore using (1.9), 2 . 3 and 2 3 are modified 푒-Perfect numbers. Also both are perfect squares therefore, these are geometric numbers too. Theorem 2.10 All almost perfect numbers which are even power of 2 are geometric numbers too. Proof: Let 푛 = 2푟 , where 푟 is an even positive integer. By the definition of almost perfect numbers introduced in Mladen V. Vassilev Missana and Krassimir T. Atanassov [6], if 푛 is even power of 2, then it is almost perfect number. Now it is clear by theorem 2.8 that 푛 is a geometric number. Preposition 2 푛 = 36 is a positive integer which is 푒-Perfect, 푒 Harmonic of type 1, 푒 Harmonic of type 2, modified 푒-Perfect and geometric numbers too. 2 2 2 2 2 2 2 2 Proof: Now 푛 = 36 = 2 . 3 . Since, 휎푒 2 . 3 = 2. 3 + 2 . 3 + 2 . 3 + 2.3 = 72, 2 2 2 2 therefore using (12) 푛 is a 푒-Perfect number. Now 푑푒 2 . 3 = 4 and 휎푒 2 . 3 = 72.

350 Shikha Yadav and Surendra Yadav

2 2 Using (1.6) 푛 is 푒-Harmonic of type 1. Since, 푝푒 2 . 3 = 2 + 1 3 + 1 = 12 therefore, using (17) 푛 is 푒 -Harmonic of type 2. Using (9), 푛 is also modified 푒-Perfect number. Since 푛 = 36 is a perfect square number therefore using (1.11) 푛 is also a geometric number. Theorem 2.11 All unitary perfect numbers are unitary harmonic too.

훼1 훼2 훼푟 Proof: Let 푛 = 푝1 .푝2 … … 푝푟 , 훼푖 ≥ 1 be an unitary perfect number. Using (14) we have 푑∗ 푛 = 2푟 and 휎∗ 푛 = 2푛. So, 휎∗ 푛 /푛. 푑∗ 푛 . Hence 푛 is an unitary harmonic number. Theorem 2.12 If 푛 = 푝2훼 with 훼 ≥ 1 and 푝 is prime then 푛 is an unitary geometric number. Proof: Let 푛 = 푝2훼 , using (14), (15), (16), (17), we have 푑∗ 푛 = 2 and 푅∗ 푛 = (푝2훼 )2/2. 1 So, 퐺∗ 푛 = (푝2훼 )2 = 푝훼 = 푛 . Conclusion A new perfect number called multiplicatively perfect number (also called M-perfect) is discussed. We have also studied about abundant, deficient, harmonic and unitary analogue of harmonic numbers. Acknowledgement: Authors are thankful to the Referee for valuable suggestions. References [1] Cohen, G.L. and Deng. M. (1998). On a generalization of Ore’s harmonic numbers, Nieuw Arch. Wiskunde. 16, 3, 161-172. [2] Cohen, G.L. and Sorli, R.M. (1998). Harmonic seeds, Fib. Quart; 36, 386-390. [3] Sandor, J. and Egri., E. (2007). Arithmetical functions in algebra, geometry and analysis, Advanced studies in contemporary Mathematics, 14, 2, 163-213. [4] Sandor, J. (2001). On multiplicatively perfect numbers, J. ineq. pure and appl. Math. 2, 1, article 3, 1-6. [5] Straus, E.G. and Subbarao, M.V. (1974). On exponential divisors, Duke. Math. J. 41, 465-471. [6] Vassilev-Missana Mladen V. and Atanassov. T. Krassimir, (2007). A new point of view on perfect and other similar numbers, Advanced Studies in contemporary mathematics 15, No. 2, 153-169.