A Counting Function for the Sequence of Perfect Powers M A Nyblom Scho ol of Mathematics and Geospatial Science RMIT University GPO Box V Melb ourne Victorial Australia email michaelnyblomrmiteduau Intro duction n A natural numb er of the form m where m is a p ositiveinteger and n is called a p erfect power Unsolved problems concerning the set of p erfect p owers ab ound throughout muchof numb er theory The most famous of these is known as the Catalan conjecture which states that the only p erfect p owers which dier by unity are the integers and It is of interest to note that this particular problem has only recently b een solved using rather deep results from the theory of cyclotomic elds see The set of p erfect p owers can naturally b e arranged into an increasing sequence of distinct integers in which those p erfect powers expressible with dieren texponents are treated as a single element of the sequence The rst few terms of this sequence of p erfect p owers without duplication are and is listed in the OnLine Encyclop edia of Integer Sequences under Sloane A The sequence in has many prop erties one b eing that the innite sum of its recipro cals is convergent see a clear indication of the scarcity of the p erfect powers amongst the set of natural numb ers This latter fact is naturally reected in the well known result that the sequence of p erfect p owers has zero asymptotic densitythat is if N x denotes the number of elements of less than a p ositivereal x then lim N xx In view of this result x one may question what is the precise nature of the growth rate of the counting function N x in particular can an asymptotic estimate for N x b e found We shall establish sucha p x as x distributional result for the sequence of p erfect p owers by proving that N x As will b e seen this asymptotic formula can b e interpreted as stating that the p erfect squares dominate the count of the sequence elements in as x To contrast the main result we shall in addition develop a closedform expression for N x using elementary sieve metho ds As will be seen this formula is some what reminiscent to Legendres counting function for p x x In what follows we denote the integer part of the number of primes in the interval x by bxc An Asymptotic Formula To help establish the main results of this pap er we shall rst need to formally intro duce the following family of sets Denition Suppose x and n N nf g thenletA x denote the set of perfect powers n n n having exponent n and which are less than or equal to x that is A xfk k N k xg n Wenow establish the asymptotic formula for the counting function N x Theorem If N x denotes the number of sequence elements of that are less than or p equal to x then N x x as x Pro of The rst step of the argument will b e to obtain upp er and lower functional b ounds for N x Assuming without loss of generality that x observe A x for each n n N nf g but for n suciently large A xnfg Dening the auxiliary function n M xmaxfn N nf g A xnfg g we clearly see M x as A xnfg and n S M x that N x is equal to the number of elements of the set A A x Furthermore from n n blog xc blog xc the inequality x it is immediately deduced that M x blog xc blog xc Since for large x the family of sets fA xg are not mutually disjoint it follows that n n blog xc X N xjAj jA xj n n and since A x A one also has jA xjjAj N x n n Now as A x there must exist a largest integer m such that m xm n p n xm that By taking the nth ro ot through the previous inequalitywe deduce m p p n n is m b xc and so A xmust contain b xc elements Consequently and together n yields that blog xc X p p n xcN x b xc b n Using the upp er and lower b ounds in we can establish required the asymptotic estimate p x observe for large x the following train of inequalities for N x as follows Dividing by p p p p p blog xc blog xc n n X X b xc N x b xc xc b xc x b p p p p p p x x x x x x n n p p blog xc X xc x b p p x x n p b blog xc xc p p x x Via an application of LHopitals rule it is easily seen that log x blog xc p p x x as x moreover by recalling lim bxcx we nally deduce from that x p N x x asx p Remark Since the number of perfect squares less than or equal to x is given by b xc p p xc x we can interpret Theorem as stating that the perfect squares dominate and as b the count of the sequence elements of as x An Exact Formula One of the earliest known sieve metho ds was a simple eective pro cedure for nding all prime numb ers up to a certain bound x This pro cedure whichinvolves the systematic deletion of p all multiples of primes less than or equal to x was captured succinctly by Legendre using a theoretical analog of the sifting pro cess known to day as the InclusionExclusion Principal to study the prime counting function x jfp x p a prime gj His metho d led to an p exact formula for the number of primes in the interval x x in particular if denotes the Mobius function then k j X p x x d x d djP x Q p where the sum is taken over all divisors of P p see pg In this section we x p x shall employ the same elementary sieve metho d of Legendre to establish an exact formula for the counting function N x which is similar in form to We b egin with a technical lemma for the sets of Denition Lemma For any set of m positive integers fn n g al l greater than unity m m A xA x n n n i m i where n n denotes the least common multiple of the m integers n n m m g Pro of We begin by demonstrating that A x A xA x for any n m N nf n m nm which is the base step of our inductive argument Nowsincenjn mandmjn manynumber nm of the form k where k N can b e rewritten as a p erfect p ower having an exp onent n and n m m thus A x A x A x Let s A x A x with s then s k k n m n m nm nm for some k k N nf g W e have to pro duce a k N nf g such that s k As n m r k k both k and k must have the same prime divisors Writing k p p p r r n m and k p p p we deduce from the equality k k that n m for each r i i i r Consequently njn and mjn and so n n m for some N Thus i i i i i r nm s k where k p p p which establishes that A x A x A x r n m nm Now supp ose for m the set identity in holds for an arbitrary set of m p ositiveintegers n n n n n n observe fn n g all greater than unity Then as m m m m from the inductive assumption and the base step that m m A x A x A x A x A x n n n n n n m m i i m i i A x n n n m m A x n n m Hence holds for m arbitrary p ositiv eintegers greater than unity and so the result is established by the principal of mathematical induction Theorem If x then the counting function for the sequence in is given by the explicit expression X d N xbxc dbx c djP x Q where the sum is taken over al l divisors of P p x pblog xc Pro of We b egin by establishing a slight reformulation for the set A of Theorem Recall blog xc ing that A A x we claim if p p are the rst m primes less than or equal n m n to blog xc then in fact A B where m x B A p r r x is included in the The inclusion B A follows automatically by denition as eachsetA p k union of sets which form A To establish the reverse inclusion A B rst observe that as p p represent the complete list of primes less than or equal to blog xc every integer m n f blog xcg must b e divisible by at least one of these primes since otherwise by log xc and the fundamental theorem of arithmetic n would be divisible by a prime p b n so n blog xc a contradiction Consequentlyifgiven any s A x then s k and one n p r x and so A may write n p for some r f mg and N Thus s k p r r every elementofA is contained in the set B m Now N xjAj jB j and since for x large the family of sets fA x g are not mutually p i i disjointwe deduce from an application of the InclusionExclusion Principal applied to the set B that m X X k N x jA x A xj p p i i k i i m k k where the expression i i m indicates that the sum is taken over all ordered k k element subsets fi i g of the set f mg As the least common multiple of the k k prime num bers p p is clearly the pro duct d p p p observe from Lemma i i i i i k k d c noting here wehave again used the fact that jA x A xj jA xj bx p p d i i k Q p n that the number of elements in the set A x is b xc Dening P p we see n x pblog xc m that for each k f mg the inner summation in consists of adding terms of k d c where d p p p is a divisor of P having k distinct prime factors the form bx i i i x k k Consequently as p p p the double summation in must sum terms of the i i i k d form dbx c over all divisors dwhere d of P excluding d Finally by recalling that x we deduce that the right hand side of reduces to the right hand side of Remark An immediate consequence of Theorem is that the number of nonperfect P d c dbx powers less than or equal to x is equal to djP x Numerical Example We examine nowhow the explicit expression for N x in can b e practically implemented to compute the numb er of p erfect p owers less than or equal to a given large p ositive real x For notational convenience let the inner summation of b e denoted by k j X p p i i k S x x k i i m k p p i i k Observe that in order to evaluate each S x one must sum the terms bx c over k those subscripts i i whose values are chosen from the ordered k element subsets k m of f mg consequently the numb er of summands is Thus on rst acquaintance