Practical numbers among the binomial coefficients
aa r X i v : . [ m a t h . N T ] M a y PRACTICAL NUMBERS AMONG THEBINOMIAL COEFFICIENTS
PAOLO LEONETTI † AND CARLO SANNA ‡ Abstract. A practical number is a positive integer n such that every positiveinteger less than n can be written as a sum of distinct divisors of n . We provethat most of the binomial coefficients are practical numbers. Precisely, letting f ( n ) denote the number of binomial coefficients (cid:0) nk (cid:1) , with 0 ≤ k ≤ n , that arenot practical numbers, we show that f ( n ) < n − (log 2 − δ ) / log log n for all integers n ∈ [3 , x ], but at most O γ ( x − ( δ − γ ) / log log x ) exceptions, for all x ≥ < γ < δ < log 2. Furthermore, we prove that the central binomialcoefficient (cid:0) nn (cid:1) is a practical number for all positive integers n ≤ x but at most O ( x . ) exceptions. We also pose some questions on this topic. Introduction A practical number is a positive integer n such that every positive integer lessthan n can be written as a sum of distinct divisors of n . This property has been in-troduced by Srinivasan [19]. Estimates for the counting function of practical num-bers have been given by Hausman–Shapiro [5], Tenenbaum [20], Margenstern [9],Saias [15], and finally Weingartner [21], who proved that there are asymptotically Cx/ log x practical numbers less than x , for some constant C >
0, as previouslyconjectured by Margenstern [9]. On another direction, Melfi [11] proved thatevery positive even integer is the sum of two practical numbers, and that thereare infinitely many triples ( n, n + 2 , n + 4) of practical numbers. Also, Melfi [10]proved that in every Lucas sequence, satisfying some mild conditions, there areinfinitely many practical numbers, and Sanna [17] gave a lower bound for theircounting function.In this work, we study the binomial coefficients which are also practical num-bers. Our first result, informally, states that for almost all positive integers n Mathematics Subject Classification.
Primary: 11B65, Secondary: 11N25.
Key words and phrases. binomial coefficient; central binomial coefficient; practical number. † P. Leonetti is supported by the Austrian Science Fund (FWF), project F5512-N26. ‡ C. Sanna is supported by a postdoctoral fellowship of INdAM and is a member of theINdAM group GNSAGA.
Paolo Leonetti and Carlo Sanna there is a negligible amount of binomial coefficients (cid:0) nk (cid:1) , with 0 ≤ k ≤ n , whichare not practical. Precisely, for each positive integer n , define f ( n ) := (cid:26) ≤ k ≤ n : (cid:18) nk (cid:19) is not a practical number (cid:27) . Our first result is the following.
Theorem 1.1.
For all x ≥ and < γ < δ < log 2 , we have f ( n ) < n − (log 2 − δ ) / log log n for all integers n ∈ [3 , x ] , but at most O γ ( x − ( δ − γ ) / log log x ) exceptions. As a consequence, we obtain that as x → + ∞ almost all binomial coefficients (cid:0) nk (cid:1) , with 0 ≤ k ≤ n ≤ x , are practical numbers. Corollary 1.1.
We have X n ≤ x f ( n ) ≪ ε x − ( log 2 − ε ) / log log x , for all x ≥ and ε > . Among the binomial coefficients, the central binomial coefficients (cid:0) nn (cid:1) are ofgreat interest. In particular, several authors have studied their arithmetic anddivisibility properties, see e.g. [1, 2, 14, 16, 18].In this direction, our second result, again informally, states that almost allcentral binomial coefficients (cid:0) nn (cid:1) are practical numbers. Theorem 1.2.
For x ≥ , the central binomial coefficient (cid:0) nn (cid:1) is a practicalnumber for all positive integers n ≤ x but at most O ( x . ) exceptions. Probably, there are only finitely many positive integers n such that (cid:0) nn (cid:1) is not apractical number. By a computer search, we found only three of them below 10 ,namely n = 4 , , n is a power of 2 whose base3 representation contains only the digits 0 and 1, then it can be shown that (cid:0) nn (cid:1) is not a practical number (see Proposition 2.1 below). On the other hand, it is anopen problem to establish whether there are finitely or infinitely many powers of2 of this type [4, 6, 8, 12].We conclude by leaving two open questions. Note that since (cid:0) n (cid:1) = (cid:0) nn (cid:1) = 1, wehave 0 ≤ f ( n ) ≤ n − n . It is natural to ask when one ofthe equalities is satisfied. Question . What are the positive integers n such that f ( n ) = 0 ? Question . What are the positive integers n such that f ( n ) = n − ractical numbers among the binomial coefficients f ( n ) = 0 then n must be a power of 2, otherwisethere would exist (see Lemma 2.4 below) an odd binomial coefficient (cid:0) nk (cid:1) , with0 < k < n , and since 1 is the only odd practical number, we would have f ( n ) > f (8) = 1. Regarding Question 1.2,if n = 2 k −
1, for some positive integer k , then f ( n ) = n −
1, because all thebinomial coefficients (cid:0) nk (cid:1) , with 0 < k < n , are odd (see Lemma 2.4 below) andgreater than 1, and consequently they are not practical numbers. However, thisis not a necessary condition, since f (5) = 4. Notation.
We employ the Landau–Bachmann “Big Oh” notation O and theassociated Vinogradov symbol ≪ . In particular, any dependence of the impliedconstants is indicated with subscripts. We write p i for the i th prime number.2. Preliminaries
This section is devoted to some preliminary results needed in the later proofs.We begin with some lemmas about practical numbers.
Lemma 2.1. If m is a practical number and n ≤ m is a positive integer, then mn is a practical number.Proof. See [10, Lemma 4]. (cid:3)
Lemma 2.2. If d is a practical number and n is a positive integer divisible by d and having all prime factors not exceeding d , then n is a practical number.Proof. By hypothesis, there exist positive integers q , . . . , q k ≤ d such that n = dq · · · q k . Then, using Lemma 2.1, it follows by induction that dq · · · q m is practical for all m = 1 , . . . , k . In particular, n is practical. (cid:3) Lemma 2.3.
We have that p a · · · p a s s is a practical number, for all positive inte-gers a , . . . , a s .Proof. It follows easily by induction on s , using Lemma 2.1 and Bertrand’s pos-tulate p i +1 < p i . (cid:3) For each prime number p and for each positive integer n , put T p ( n ) := (cid:26) ≤ k ≤ n : p ∤ (cid:18) nk (cid:19)(cid:27) . We have the following formula for T p ( n ). Lemma 2.4.
Let p be a prime number and let n = s X j = 0 d j p j , d , . . . , d s ∈ { , . . . , p − } , d s = 0 , Paolo Leonetti and Carlo Sanna be the representation in base p of the positive integer n . Then we have T p ( n ) = s Y j = 0 ( d j + 1) . Proof.
See [3, Theorem 2]. (cid:3)
For each prime number p , let us define ω p := log(( p + 1) / p . The quantity ω p appears in the following upper bound for T p ( n ). Lemma 2.5.
Let p be a prime number and fix ε ∈ (0 , / ) . Then, for all x ≥ ,we have T p ( n ) < n ω p + ε for all positive integers n ≤ x but at most O ( p x − ε ) exceptions.Proof. For x ≥
1, let k be the smallest integer such that x < p k . Clearly, we have E ( x ) := (cid:8) n ≤ x : T p ( n ) ≥ n ω p + ε (cid:9) ≤ k X j = 1 (cid:8) p j − ≤ n < p j : T p ( n ) ≥ p ( j − ω p + ε ) (cid:9) . (1)Moreover, thanks to Lemma 2.4, we have X p j − < n ≤ p j T p ( n ) ≤ X ≤ d ,...,d j − < p j − Y i = 0 ( d i + 1) = p − X d = 0 ( d + 1) ! j = (cid:18) p ( p + 1)2 (cid:19) j , and consequently (cid:8) p j − ≤ n < p j : T p ( n ) ≥ p ( j − ω p + ε ) (cid:9) ≤ p ( j − ω p + ε ) X p j − < n ≤ p j T p ( n ) ≤ p ( j − ω p + ε ) (cid:18) p ( p + 1)2 (cid:19) j = p ( p + 1)2 p (1 − ε )( j − < p − ε )( j − , (2)for all positive integers j . Therefore, putting together (1) and (2), and using that ε < / , we obtain E ( x ) < k X j = 1 p − ε )( j − ≪ p − ε ) k ≤ p − ε )(log x/ log p +1) < p x − ε , (3)as desired. (cid:3) ractical numbers among the binomial coefficients Remark . The constant / in the statement of Lemma 2.5 has no particularimportance, it is only needed to justify the ≪ in (3). Any other real number lessthan 1 would be fine.For all x ≥
1, let κ ( x ) be the smallest integer k ≥ p · · · p k ≥ x . Lemma 2.6.
We have κ ( x ) ∼ log x log log x and p κ ( x ) ∼ log x, as x → ∞ .Proof. As a well-known consequence of the Prime Number Theorem, we havelog( p · · · p k ) ∼ p k ∼ k log k, (4)as k → + ∞ . Since log( p · · · p κ ( x ) − ) < log x ≤ log( p · · · p κ ( x ) ) , and κ ( x ) → + ∞ as x → + ∞ , by (4) we obtain p κ ( x ) ∼ κ ( x ) log κ ( x ) ∼ log x, which in turn implies κ ( x ) ∼ κ ( x ) log κ ( x )log κ ( x ) + log log κ ( x ) ∼ log x log log x , as desired. (cid:3) For every prime number p and every positive integer n , let β p ( n ) be the p -adicvaluation of the central binomial coefficient (cid:0) nn (cid:1) . Lemma 2.7.
For each prime p and all positive integers n , we have that β p ( n ) isequal to the number of digits of n in base p which are greater than ( p − / .Proof. The claim is a straightforward consequence of a theorem of Kummer [7]which says that, for positive integers m, n , the p -adic valuation of (cid:0) m + nn (cid:1) is equalto the number of carries in the addition m + n done in base p . (cid:3) Proposition 2.1. If n is a power of and if all the digits of n in base are equalto or , then (cid:0) nn (cid:1) is not a practical number.Proof. It follows by Lemma 2.7 that β ( n ) = 1 and β ( n ) = 0, that is, (cid:0) nn (cid:1) is aninteger of the form 12 k ±
2. However, it is known that, other than 1 and 2, everypractical number is divisible by 4 or 6, see [19]. (cid:3)
We will make use of the following result of probability theory.
Paolo Leonetti and Carlo Sanna
Lemma 2.8.
Let X be a random variable following a binomial distribution with j trials and probability of success α . Then P [ X ≤ ( α − ε ) j ] ≤ e − ε j for all ε > .Proof. See [13, Theorem 1]. (cid:3)
For each prime number p , let us define α p := 1 p (cid:24) p − (cid:25) , so that α p is the probability that a random digit in base p is greater than ( p − / Lemma 2.9.
Let p be a prime number and fix ε ∈ (0 , / ) . Then, for all x ≥ ,we have β p ( n ) > ( α p − ε ) log n log p for all positive integers n ≤ x but at most O ( px − ε / log p ) exceptions.Proof. For x ≥
1, let k be the smallest integer such that x < p k . Clearly, we have E ( x ) := (cid:26) n ≤ x : β p ( n ) ≤ ( α p − ε ) log n log p (cid:27) ≤ k X j = 1 (cid:8) p j − ≤ n < p j : β p ( n ) ≤ ( α p − ε ) j (cid:9) ≤ k X j = 1 (cid:8) ≤ n < p j : β p ( n ) ≤ ( α p − ε ) j (cid:9) . (5)Given an integer j ≥
1, let us for a moment consider n as a random variableuniformly distributed in { , . . . , p j − } . Then, the digits of n in base p are j independent random variables uniformly distributed in { , . . . , p − } . Hence, asa consequence of Lemma 2.7, we obtain that β p ( n ) follows a binomial distributionwith j trials and probability of success α p . In turn, Lemma 2.8 yields { ≤ n < p j : β p ( n ) ≤ ( α p − ε ) j } ≤ p j e − ε j . (6)Therefore, putting together (5) and (6), and using that ε < / , we get E ( x ) ≤ k X j = 1 p j e − ε j ≪ ( pe − ε ) k ≤ ( pe − ε ) log x/ log p +1 < px − ε / log p , (7)as desired. (cid:3) ractical numbers among the binomial coefficients Remark . The constant / in the statement of Lemma 2.9 has no particularimportance, it is only needed to justify the ≪ in (7). Any other real number lessthan ( log 2) / would be fine.3. Proof of Theorem 1.1
Assume x ≥ ε := δ − γ log log x + 4 log log x log x ∈ (0 , / ) . Let n be a positive integer. By Lemma 2.3 and by the definition of κ ( n ), weknow that p · · · p κ ( n ) is a practical number greater than or equal to n . Since allthe prime factors of (cid:0) nk (cid:1) are not exceeding n , Lemma 2.2 tell us that if p · · · p κ ( n ) divides (cid:0) nk (cid:1) then (cid:0) nk (cid:1) is practical. Consequently, we have f ( n ) ≤ (cid:26) ≤ k ≤ n : p · · · p κ ( n ) ∤ (cid:18) nk (cid:19)(cid:27) ≤ κ ( n ) X j = 1 T p j ( n ) . Therefore, it follows from Lemma 2.5 that f ( n ) < κ ( n ) X j = 1 n ω pj + ε , (8)for all positive integers n ≤ x but at most ≪ κ ( x ) X j = 1 p j x − ε ≪ p κ ( x ) x − ε ≪ (log x ) x − ε = x − ( δ − γ ) / log log x exceptions, where we also used Lemma 2.6.Suppose that n satisfies (8). Since ω p is a monotone increasing function of p ,we get that f ( n ) < κ ( n ) n ω pκ ( n ) + ε = n ω pκ ( n ) +log κ ( n ) / log n + ε . (9)Moreover, for n ≫ γ ω p κ ( n ) < − log 2log p κ ( n ) + 1 p κ ( n ) log p κ ( n ) < − log 2 − γ/ n , (10)and log κ ( n )log n < γ/ n , (11)where we used Lemma 2.6. Furthermore, since n ≤ x , we have ε < δ − γ/ n . (12) Paolo Leonetti and Carlo Sanna
Consequently, putting together (10), (11), and (12), we obtain ω p κ ( n ) + log κ ( n )log n + ε < − log 2 − δ log log n , which, inserted into (9), gives f ( n ) < n − (log 2 − δ ) / log log n as desired. The proof is complete.4. Proof of Corollary 1.1
Obviously, we can assume ε < log 2. Put γ := 2 ε and δ := log 2 + ε , so that0 < γ < δ < log 2. For all x ≥
3, let E ( x ) be the set of exceptional n ≤ x ofTheorem 1.1. Then we have X n ≤ x f ( n ) = X n / ∈ E ( x ) f ( n ) + X n ∈ E ( x ) f ( n ) < X n ≤ x n − (log 2 − δ ) / log log n + E ( x ) x ≪ ε x − (log 2 − δ ) / log log x + x − ( δ − γ ) / log log x ≪ x − ( log 2 − ε ) / log log x , as claimed. 5. Proof of Theorem 1.2
For the sake of notation, put s := 16 , η s := P si =1 α p i − P si =1 √ log p i , ε j := η s p log p j , for j = 1 , . . . , s . A computation shows that ε j ∈ (0 , / ) for j = 1 , . . . , s .For x ≥
1, it follows from Lemma 2.9 that s X j = 1 β p j ( n ) log p j > s X j = 1 ( α p j − ε j ) log n = log n, (13)for all positive integers n ≤ x , but at most ≪ s X j = 1 p j x − ε j / log p j ≪ x − η s < x . exceptions. Suppose that n is a positive integer satisfying (13). Then, d := s Y j = 1 p β pj ( n ) j > n. Also, by Lemma 2.3 we have that d is a practical number, and by the definitionof β p j ( n ) we have that d divides (cid:0) nn (cid:1) . Moreover, since all the prime factors of ractical numbers among the binomial coefficients (cid:0) nn (cid:1) are not exceeding 2 d , Lemma 2.2 yields that (cid:0) nn (cid:1) is practical. The proof iscomplete. Remark . A comment is in order to explain the choice of the parameters ε j inthe proof of Theorem 1.2. Given a positive integer s , one could fix some primenumbers q < · · · < q s and some real numbers ε , . . . , ε s ∈ (0 , / ) such that q · · · q s is a practical number and P sj =1 ( α q j − ε j ) ≥
1. Everything would proceedsimilarly, with an estimate of the number of exceptions given by O (cid:16) x max { − ε / log q ,..., − ε s / log q s } (cid:17) . To minimize the exponent of x , the optimal choice for ε j is ε j = η s ( q , . . . , q s ) p log q j , η s ( q , . . . , q s ) := P si =1 α q i − P si =1 p log q j , for j = 1 , . . . , s , which gives the estimate O (cid:16) x − η s ( q ,...,q s ) (cid:17) . Since α p = + O ( p ) for each prime number p , we get that η s ( q , . . . , q s ) is max-imized when q j = p j , for j = 1 , . . . , s , and that η s ( p , . . . , p s ) → s → + ∞ .Lastly, some numeratical computations verify that the maximum of η s ( p , . . . , p s )is reached for s = 16. References
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