Pure Mathematcal Scences, Vol. 4, 205, no. 3, 99-03 HIKARI Ltd, www.m-har.com http://dx.do.org/0.2988/pms.205.4923 A Lower Bound for τn of Any -Perfect Numbers Keneth Adran P. Dagal Department of Mathematcs Far Eastern Unversty Manla, Phlppnes Copyrght c 204 Keneth Adran P. Dagal. Ths artcle s dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal wor s properly cted. Abstract A natural number n s sad to be - perfect number f σn = n for some nteger 2. In ths paper, we wll provde a lower bound for τn of any -perfect numbers. The lower bound for τn wll help n determnng f the number s a possble -perfect or not.for example, for all n where τn < 40, 427, 833, 596, the number n can never be a -perfect number wth 25. Mathematcs Subject Classfcaton: 0A Keywords: -perfect numbers, abundancy ndex Introducton The number of dvsors of any natural number n, n = m = pα, s gven by the formula τn = d n = m = α + and the sum of the dvsors of any natural number n s defned as σn = d n d where d s a dvsor of n. In studyng perfect numbers, the abundancy ndex, denoted by In,s defned as In = σn. And lastly,a natural number n s sad to be -perfect number f n σn = n for some nteger 2. If n s -perfect, then In = d n =. d On the other hand, we now that the nth harmonc number denoted by H n s defned as H n = n =. Clearly, In H n for all natural numbers n.
00 Keneth Adran P. Dagal 2 Prelmnary Results Let us frst consder some lemmas. Lemma 2.. For N, the nequalty + holds. + 2 + + + Proof 2.2. Consder frst the nequalty + + + + + By bnomal expanson on the RHS of the nequalty, we have + = + Clearly, =0 0 =2 + = + + +. + =2 + Addng both sdes by +, we arrve on the desred nequalty. On the other + hand, consder the nequalty + + + 2 + Rasng both sdes by + 2, we get + + 2 Snce the lm x + +2 + + + x = e x +2 + x + y+. x y lm y + + y+ = e y from below and from above respectvely,by usng the bnomal theorem and the power seres expanson of e, then that proves the nequalty. Lemma 2.3. The nequalty holds. =2 < n =.
A lower bound for τn of any -perfect numbers 0 Proof 2.4. By lemma, By some manpulatons, + + + 2 + + 2 + Thus, we get Now, we consder the nequalty + ln < e e Therefore, + + + 2 + + 2 + + 2 + + + + < e + + + ln < ln + ln < = + + n < e + ln <. = The other nequalty s left as an exercse. Remar 2.5. In fact,the nequalty =2 n < x dx = + 2 +. = + appeared as a problem n the boo of Rosen. Ths was noted here to be able to mae the followng connectons. The prevous lemma can be wrtten as H n < H n γ + ɛ < H n
02 Keneth Adran P. Dagal where γ s the Euler- Mascheron constant, defned as γ = lm H n lnn n + and ɛ can be seen as the error term. For more detals about Euler- Mascheron constant, you may loo at the paper of Lagaras. From ths nequalty, we can have a bound for γ. ɛ < γ < ɛ As n +, ɛ 0 and that wll gve us 0 < γ <. In fact, γ = 0.577256649053286060652... see Sloane s A00620 at OEIS.org 3 Results and Dscusson Theorem 3.. For postve ntegers, = where for every and j, j and for all and +, < +. Proof 3.2. It should be noted that equalty holds f =. Now suppose that there exsts. Ths would mean that n the set S = {, 2, 3,..., n}, there s / S. Thus, > n. Now, we have s such that = < n < j for all j S such that j. Addng all unt fractons j j =, we get + j + j j j= j j= and thus, = = for j and Suppose that s are not just any random natural numbers but rather all n and the n n the n = wll be replaced wth τn. From ths, we can rewrte the above nequalty as = In = d n τn d = H τn d = d n
A lower bound for τn of any -perfect numbers 03 Theorem 3.3 A Lower bound of τn. For large n,n can be a -perfect number f the property e γ < τn s satsfed. Proof 3.4. It was already establshed that < H τn = lnτn + γ Then, γ < lnτn e γ < τn Acnowledgements. The author would le to than Jose Arnaldo Drs,for nsprng hm n dong ths research, Calvn Ln, for the advce n provng Lemma, and Solomon Olayta, for useful conversaton about unt fractons and Immanuel San Dego, for the support and motvaton. References [] Drs, Jose Arnaldo B. The Abundancy Index of dvsors of odd perfect numbers, Journal of Integer Sequences.,5 Sept 202, Artcle 2.4.4, https://cs.uwaterloo.ca/journals/jis/vol5/drs/drs8.html, ISSN 530-7638 [2] Lagaras, Jeffrey C. Euler s constant: Euler s wor and modern developments, Bulletn of the Amercan Mathematcal Socety, 50 203, 527-628. http://dx.do.org/0.090/s0273-0979-203-0423-x [3] Rosen, Kenneth H.2008. Dscrete Mathematcs and Its Applcaton: 6th ed.mcgraw-hll Educaton Asa. pp 92, problem no. 60. [4] N. J. A. Sloane, The On-Lne Encyclopeda of Integer Sequences, Sequence A00620. see: http://oes.org/a00620, http://dx.do.org/0.007/978-3-540-73086-6 2 [5] N. J. A. Sloane, The On-Lne Encyclopeda of Integer Sequences, Sequence A004080. see: http://oes.org/a004080, http://dx.do.org/0.007/978-3-540-73086-6 2 Receved: September 28, 204; Publshed: June 6, 205