A Note on the Distribution of Numbers with a Maximum (Minimum) Fixed Prime Factor

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1 International Mathematical Forum, Vol. 7, 2012, no. 13, A Note on the Distribution of Numbers with a Maximum Minimum) Fixed Prime Factor Rafael Jakimczuk División Matemática, Universidad Nacional de Luján Buenos Aires, Argentina jakimczu@mail.unlu.edu.ar Abstract In this note we study the distribution of numbers with a maximum minimum) fixed prime factor in their prime factorization. Mathematics Subject Classification: 11A99, 11B99 Keywords: Numbers with a maximum minimum) prime factor fixed 1 Preliminary Results Let us consider the sequence of all positive integers whose factorization is of the form q s q s k k where s i 0 i = 1, 2,...,k) and q 1,...q k k 2) are distinct primes fixed. Let ψx) denote the number of these integers not exceeding x. We need the following theorem. Theorem 1.1 The following formula holds ψx) = ln k x 1 1 k!lnq 1 ln q k 2 k 1)! ln q 1 lnq k ln q 1 ln q k ln k1 x oln k1 x). 1) Proof. See [2]. The following theorem is sometimes called either the principle of inclusionexclusion or the principle of cross-classification. We now enunciate the principle. Theorem 1.2 Let S be a set of N distinct elements, and let S 1,...,S r be arbitrary subsets of S containing N 1,...,N r elements, respectively. For 1 i < j <... < l r, let S ij...l be the intersection of S i,s j,...,s l and let

2 616 R. Jakimczuk N ij...l be the number of elements of S ij...l. Then the number K of elements of S not in any of S 1,...,S r is K = N N i N ij N ijk 1) r N 12...r. 2) 1 i r 1 i<j r 1 i<j<k r In particular if S = r S i then K =0and consequently N = N i N ij N ijk 1) r1 N 12...r. 3) 1 i r 1 i<j r 1 i<j<k r Proof. See, for example, either [1, page 233] or [3, page 84]. We also need the binomial formula, a b) k = k i=0 ) k a ki b i, 4) i and the following well-known limit see [1, chapter XXII]) ) n lim 1 1pi =0, 5) n where p i is the i-th prime number. The former limit is a weak consequence of the following Mertens s Theorem. Theorem 1.3 Mertens) The following asymptotic formula holds 1 1 ) eγ p log x, p x where γ is Euler s constant and p denotes a positive prime number. Proof. See [1, chapter XXII, Theorem 429]. In this note as usual). denotes the integer-part function. Note that 2 Main Results 0 x x < 1. In this section p n denotes the n-th prime number. Then p 1 =2,p 2 =3,p 3 = 5,p 4 =7,p 5 =11,... Let us consider the sequence of all positive integers whose prime factorization is of the form p s 1 1 p s p s k k where s i 0 i =1, 2,...,k1) and s k 1. That is, the sequence of all positive integers such that the maximum prime factor in their prime factorization is p k k 2). Let αx) be the number of these numbers that do not exceed x. We have the following theorem.

3 Numbers with a maximum minimum) prime factor 617 Theorem 2.1 The following asymptotic formula holds ln k x αx) = 1 1 ln p 1 lnp 2 ln p k ln k1 xoln k1 x) k!lnp 1 ln p k 2 k 1)! ln p 1 ln p k 6) Proof. We have see 1)) ) ) x x ln ) k x p αx) = ψ = ψ = k p k p k k!lnp 1 ln p k 1 ) 1 ln p 1 lnp 2 lnp k x ln k1 2 k 1)! ln p 1 ln p k p k )) o ln k1 = xpk ln x ln p k) k k!lnp 1 ln p k 1 1 ln p 1 lnp 2 lnp k ln x ln p k ) k1 2 k 1)! ln p 1 ln p k o ln x ln p k ) k1). 7) On the other hand we have see 4)) ln x ln p k ) k =ln k x k ln p k ln k1 x o ln k1 x ) 8) ln x ln p k ) k1 =ln k1 x o ln k1 x ) 9) Substituting 8) and 9) into 7) we obtain 6). The theorem is proved. Let us consider the set β ph of all positive integers whose prime factorization is of the form p s h h p s h1 h1... where s i 0 i = h 1,h2,...) and s h 1. That is, the set β ph of all positive integers such that the minimum prime factor in their prime factorization is p h. Note that if i j then the sets β pi and β pj are disjoints. On the other hand β pi = N {1}, where N is the set of all positive integers. That is, the sets β ph h =1, 2,...) are a partition of N {1}. The density of N {1} is 1. We shall prove that the set β ph h =1, 2,...) has positive density D ph and that the sum of the infinite positive densities is 1. That is, D ph = 1. Consequently the sum of the densities of the infinite sets β ph equals the density of the union of these infinite sets. Let β ph x) be the number of numbers in the set β ph that do not exceed x and let δ ph x) be the set of numbers in the set β ph that do not exceed x. Then the number of elements in the set δ ph x) isβ ph x). We have the following theorem. Theorem 2.2 The following asymptotic formula holds h1 β ph x) = 1 1 )) 1 x O1). 10) p i p h

4 618 R. Jakimczuk Consequently these numbers have positive density h1 D ph = 1 1 )) 1. 11) p i p h Proof. We have see 2)) x β ph x) = x x x p h 1 i h1 p h p i 1 i<j h1 p h p i p j 1 i<j<k h1 p h p i p j p k 1) h1 x = x x x p h p 1 p h1 p h 1 i h1 p h p i 1 i<j h1 p h p i p j x 1) h1 x O1) 1 i<j<k h1 p h p i p j p k p h p 1 p h1 = x h1 )) 1 1pi O1). 12) p h The theorem is proved. We now prove that the sum of the infinite positive densities is 1. We give two proof of this theorem. Theorem 2.3 The following formula holds, h1 )) D ph = 1 1pi = 1ph ) ) 1 1 ) ) 1 1 ) =1. 13) 2 3 5) 7 First proof. Without difficulty can be proved by mathematical induction the following equality n n h1 D ph = 1 1 )) ) 1 n =1 1 1pi. 14) p i p h Equations 14) and 5) give 13). The theorem is proved. Second proof. Let χ ph x) be the set of numbers multiples of p h that do not exceed x. Let Nx) be the number of elements in n χ ph x). We have see 3)) Nx) = x x x p i p i p j p i p j p k 1 i n 1) n1 1 i<j n x p 1 p 2 p n = 1 i n 1 i<j<k n x p i 1) n1 x p 1 p 2 p n O1) = 1 i<j n n 1 x p i p j )) 1 1pi 1 i<j<k n x p i p j p k x O1).

5 Numbers with a maximum minimum) prime factor 619 Consequently ) Nx) n lim =1 1 1pi. 15) x x On the other hand, we have the equality n n χ ph x) = δ ph x). Note that if i j then the sets δ pi x) and δ pj x) are disjoints. Consequently we have see 10)) Therefore n n h1 Nx) = β ph x) = 1 1 )) 1 x O1). p i p h Nx) lim x x = n h1 1 1 p i )) 1 p h. 16) Equations 15) and 16) give 14). Finally, equations 14) and 5) give 13). The theorem is proved. To finish, we prove the following theorem. Theorem 2.4 The following asymptotic formula holds D ph eγ h log 2 h 17) Proof. We have see 11) and Mertens s Theorem with x = p h1 ) D ph = h1 1 1 )) 1 e γ 1 = fh), p i p h log p h1 p h where fh) 1. Now Prime Number Theorem) we have p h h log h and log p h1 log p h log h. This proves 17). The theorem is proved. ACKNOWLEDGEMENTS. The author is very grateful to Universidad Nacional de Luján. References [1] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fourth Edition, Oxford, 1960.

6 620 R. Jakimczuk [2] R. Jakimczuk, Integers of the form p s 1 1 p s 2 2 p s k k where p 1,p 2,...,p k are primes fixed, Int. J. Contemp. Math. Sciences, ), [3] W. J. LeVeque, Topics in Number Theory, Addison-Wesley, Received: August, 2011