The Biharmonic Mean of the Divisors of N

THE BIHARMONIC MEAN MARCO ABRATE, STEFANO BARBERO, UMBERTO CERRUTI, NADIR MURRU University of Turin, Department of Mathematics Via Carlo Alberto 10, Turin, Italy [email protected], [email protected] [email protected], [email protected] Abstract We briefly describe some well–known means and their properties, focusing on the relationship with integer sequences. In particular, the harmonic numbers, de- riving from the harmonic mean, motivate the definition of a new kind of mean that we call the biharmonic mean. The biharmonic mean allows to introduce the biharmonic numbers, providing a new characterization for primes. Moreover, we highlight some interesting divisibility properties and we characterize the semi–prime biharmonic numbers showing their relationship with linear recurrent sequences that solve certain Diophantine equations. Keyword: arithmetic mean, divisibility, geometric mean, harmonic mean, harmonic numbers, integer sequences. AMS Subject Classification: 11N80, 26E60. 1 INTRODUCTION The need to explore the Nature and establish from direct observations its rules, encouraged the ancient thinkers in finding appropriate mathematical tools, able to extrapolate numerical data. The arithmetic mean is one of the arXiv:1601.03081v1 [math.NT] 12 Jan 2016 oldest quantities introduced for this purpose, in order to find an unique ap- proximate value of some physical quantity from the empirical data. It has been probably used for the first time in the third century B.C., by the ancient Babylonian astronomers, in their studies on the positions and motions of ce- lestial bodies. The mathematical relevance of the arithmetic mean has been enhanced by the Greek astronomer Hipparchus (190–120 B.C.). Some other Greek mathematicians, following the Pythagoric ideals, have also introduced and rigorously defined further kinds of means. For example, Archytas (428– 360 B.C.) named the harmonic mean and used it in the theory of music and in the algorithms for doubling the cube. His disciple Eudoxus (408–355 B.C.) 1 Biharmonic mean introduced the contraharmonic mean in his studies on proportions. At the same time as the practical use of many numerical means in various sciences, a deep exploration of their arithmetic and geometric properties took place dur- ing the centuries. The book of Bullen [2] is a classical reference for a good survey about the various kinds of means and their history. In this paper, we especially focus our attention on some arithmetic aspects related to the most used means in many fields of mathematics. We also define a new kind of mean, showing how it also allows to give a new characterization for the prime numbers. We start recalling the classical definitions and properties of the involved means. Definition 1. Let a1, a2, . , at be t positive real numbers, then we define their arithmetic mean • a1 + a2 + + at (a , a , . , at)= · · · ; (1) A 1 2 t geometric mean • t (a , a , . , at)= √a a at; (2) G 1 2 1 2 · · · harmonic mean • − 1 1 1 1 1 (a1, a2, . , at)= + + + ; (3) H t a a · · · at 1 2 contraharmonic mean • 2 2 2 a1 + a2 + + at (a1, a2, . , at)= · · · . (4) C a + a + + at 1 2 · · · We have the well–known inequalities (a , a , . , at) (a , a , . , at) (a , a , . , at) (a , a , . , at). H 1 2 ≤ G 1 2 ≤A 1 2 ≤C 1 2 A very interesting problem is to determine whether at least one of these equalities holds. If we have only two positive real numbers a and b, we always obtain the following relations: (a, b)= ( (a, b), (a, b)), (5) A A H C 2 Biharmonic mean (a, b)= ( (a, b), (a, b)). (6) G G H A In 1948 Oystein Ore ([4], [5]) introduced the idea of the harmonic number finding related properties and a quite surprising answer to this question. He evaluated the four means of all the divisors of a positive integer n. Let us denote the set of divisors of n as D(n)= d , d ,...,dt . { 1 2 } For the sake of simplicity we pose A(n) = (d , d ,...,dt), A 1 2 G(n) = (d , d ,...,dt), G 1 2 H(n) = (d , d ,...,dt), H 1 2 C(n) = (d , d ,...,dt). C 1 2 Recalling that the divisor function is t x σx(n)= di , x N (7) i ∈ X=1 the first immediate result of Ore can be summarized in the following theorem. Theorem 1. σ (n) A(n)= 1 , (8) σ0(n) G(n)= √n, (9) nσ (n) H(n)= 0 , (10) σ1(n) σ (n) C(n)= 2 . (11) σ1(n) Proof. We give a proof only for equalities (10) and (9) since (8) and (11) clearly arise from (1) and (4), respectively. From equation (3) clearly − 1 1 1 1 1 H(n)= + + + t d d · · · dt 1 2 1 1 1 and the sum d + d + + d , when reduced to the least common denominator 1 2 · · · t n, gives as numerator σ1(n) (the sum of all divisors of n). Moreover, since 3 Biharmonic mean t = σ0(n) (the number of all divisors of n), relation (10) easily follows. Now, to prove equation (9), we start from (2) σ0(n) t σ (n) G(n)= d d dt = 0 di 1 2 v · · · u i u Y=1 p t distinguishing two cases: n = m2 or n = m2. When n = m2, n has an even 6 n 6 σ (n) di i ,..., 0 number of divisors, so we multiply by di for all = 1 2 , finding σ n σ (n) 0( ) 0 2 i=1 di = n 2 . On the other hand, if n = m , then t = σ0(n) is odd. n Similarly, we multiply di by , but in this case we can do this only when Q di di = m, finding 6 σ0(n) σ n − σ n − 0( ) 1 2 0( ) 1 σ (n) di = n 2 m = (m ) 2 m = m 0 . i Y=1 Thus σ0(n) σ n σ (n) 0( ) σ n G(n)= 0 di = m 0( ) = m = √n. v u i u Y=1 p t A straightforward consequence, observed by Ore in [4], shows that a similar equality to (6) holds taking into account elements of D(n). Corollary 1. For any positive integer n G(n)= (H(n), A(n)). (12) G Proof. By previous theorem, we clearly obtain nσ (n) σ (n) (H(n), A(n)) = H(n) A(n)= 0 1 = √n = G(n). G · s σ1(n) σ0(n) p Equality (12) can be interpreted also as a formal identity. For example when n = p2q, with p and q primes, D(n)= 1,p,q,pq,p2,p2q and, by (12), { } we have (1,p,q,pq,p2,p2q)= ( (1,p,q,pq,p2,p2q), (1,p,q,pq,p2,p2q)). (13) G G H A 4 Biharmonic mean The astonishing fact is that this equality also holds substituting p and q with any other couple of positive real numbers. For example, if we use p = √2 and q = π, we obtain the identity (1, √2, π, √2 π, 2, 2π)= ( (1, √2, π, √2 π, 2, 2π), (1, √2, π, √2 π, 2, 2π)). G G H A which is not so immediate. As previously observed, for randomly chosen positive distinct real numbers a1, . , at the equality (a , a , . , at)= ( (a , a , . , at), (a , a , . , at)), G 1 2 G H 1 2 A 1 2 is false. The second question is pretty natural: when do the means of the divisors of an integer n also give a result which is an integer? The case of G(n) is not so interesting because G(n) is an integer if and only if n is a square. The integers n for which A(n) is an integer form the sequence A003601 in OEIS [6]: 1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37,... (14) These integers are the so–called arithmetic numbers. Moreover, every integer n giving an integer value for C(n) belongs to the sequence A020487: 1, 4, 9, 16, 20, 25, 36, 49, 50, 64, 81, 100, 117, 121, 144, 169, 180, 196, 200, 225,.... The most interesting case is related to the harmonic mean. Ore provided the following definition. Definition 2. A positive integer n is called a harmonic (divisor) number (or Ore number) if H(n) is an integer. The first harmonic numbers are 1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190, 18600, 18620, 27846, 30240,... and they form the sequence A001599. The corresponding values of H(n) are listed in A001600: 1, 2, 3, 5, 6, 5, 8, 9, 11, 10, 7, 15, 15, 14, 17, 24,.... Ore also proved that all perfect numbers are harmonic numbers. In the next section, moving from the above beautiful properties, we will define a new kind of mean which we will call a biharmonic mean. We will also define the biharmonic numbers, which provide a new characterization for prime numbers. Moreover, the composite biharmonic numbers will lead to the study of interesting divisibility properties, involving consecutive terms of linear recurrent sequences and solutions of Diophantine equations. 5 Biharmonic mean 2 BIHARMONIC MEAN AND BIHARMONIC NUMBERS Definition 3. For positive real numbers a1, a2, . , at, we define the biharmonic mean as (a1, a2, . , at)+ (a1, a2, . , at) (a , a , . , at)= ( (a , . , at), (a , . , at)) = H C B 1 2 A H 1 C 1 2 which corresponds to the arithmetic mean of the harmonic and contraharmonic means of ai’s.

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