Tau Numbers

Having just posted about anti-tau numbers, I realised that I hadn't yet made a dedicated post about tau numbers and in fact only mentioned them briefly in a post titled Arithmetic Numbers. In this post, I'll address that deficiency. 

Wikipedia has the following definition:

A refactorable number or tau number is an integer \(n\) that is divisible by the count of its divisors, or to put it algebraically, \(n\) is such that \( \tau (n) \mid n \). The first few refactorable numbers are listed in OEIS A033950 as:

1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225, 228, 232, 240, 248, 252, 276, 288, 296, ...

For example, 18 has 6 divisors (1 and 18, 2 and 9, 3 and 6) and is divisible by 6. There are infinitely many refactorable numbers.

Up to 100,000, there are 5257 tau numbers representing 5.257% of the range. However, the article points out that these numbers have a natural density of zero. Another Wikipedia article explains what is meant by this term:

In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the desired subset when combing through the interval [1, \(n\)] as \(n\) grows large.

Intuitively, it is thought that there are more positive integers than perfect squares, since every perfect square is already positive, and many other positive integers exist besides. However, the set of positive integers is not in fact larger than the set of perfect squares: both sets are infinite and countable and can therefore be put in one-to-one correspondence. Nevertheless if one goes through the natural numbers, the squares become increasingly scarce. The notion of natural density makes this intuition precise for many, but not all, subsets of the naturals (see Schnirelmann density, which is similar to natural density but defined for all subsets of \( \mathbb{N} \).

If an integer is randomly selected from the interval [1, \(n\)], then the probability that it belongs to A is the ratio of the number of elements of A in [1, \(n\)] to the total number of elements in [1,\( n\)]. If this probability tends to some limit as \(n\) tends to infinity, then this limit is referred to as the asymptotic density of A. This notion can be understood as a kind of probability of choosing a number from the set A. Indeed, the asymptotic density (as well as some other types of densities) is studied in probabilistic number theory.

It has been proven that there are no three consecutive integers that are all tau numbers. They can appear in pairs however, although not often. For example, up to 100,000, there are only 13 such pairs. These are:

(1, 2), (8, 9), (1520, 1521), (50624, 50625), (62000, 62001), (103040, 103041), (199808, 199809), (221840, 221841), (269360, 269361), (463760, 463761), (690560, 690561), (848240, 848241), (986048, 986049)

Take the last pair as an example:

\(986048 = 2^6 \times 7 \times 31 \times 71\) with \(56\) divisors such that \( 56 \mid 986048= 17608\)

\(986049 = 3^2 \times 331^2\) with \(9\) divisors such that \(9 \mid 986049= 109561\)

Whether there are an infinite number of such pairs is not known. Numbers Aplenty states that the smallest Pythagorean triple of tau numbers is (40, 96,104) which is not a primitive triple because it is a multiple of (5, 12, 13).  No one knows if there is a primitive triple.

Up to one million, there are 60 palindromic tau numbers. They are:

[1, 2, 8, 9, 88, 232, 252, 424, 444, 636, 808, 828, 2772, 4224, 12321, 21512, 21612, 23032, 23832, 24642, 25352, 25452, 27372, 29292, 40104, 40904, 42324, 42424, 42624, 44244, 46164, 46264, 46464, 48084, 48384, 48584, 48684, 61416, 61816, 63036, 63636, 65856, 67476, 67576, 69396, 69896, 80508, 82428, 84248, 84948, 86168, 86868, 88188, 88488, 216612, 270072, 423324, 426624, 468864, 486684]

Again, up to one million, there are also 2731 non-palindromic tau numbers whose reversals are also tau numbers. The first such number is 80 with reversal 8. The initial members of this sequence are:

80, 276, 288, 468, 480, 672, 864, 880, 882, 1440, 1656, 2000, 2025, 2148, 2160, 2176, 2178, 2196, 2320, 2388, 2700, 2988, 4044, 4050, 4068, 4080, 4240, 4284, 4404, 4668, 4824, 4856, 4860, 4896, 5202, 5220, 6561, 6584, 6712, 6720, 6912, 6984, 8080, 8100, 8412, 8604, 8649, 8664, 8712, 8832, 8892, 9468, 10000, ... permalink

It should be noted that \( \tau(n)=\sigma(n,0)\) and so this function can be used as an alternative to len(divisors(\(n\))) in any calculations.

There's an interesting history associated with the term refactorable number. To quote again from the Wikipedia article:

First defined by Curtis Cooper and Robert E. Kennedy where they showed that the tau numbers have natural density zero, they were later rediscovered by Simon Colton using a computer program he had made which invents and judges definitions from a variety of areas of mathematics such as number theory and graph theory. Colton called such numbers "refactorable". While computer programs had discovered proofs before, this discovery was one of the first times that a computer program had discovered a new or previously obscure idea. Colton proved many results about refactorable numbers, showing that there were infinitely many and proving a variety of congruence restrictions on their distribution. Colton was only later alerted that Kennedy and Cooper had previously investigated the topic.

Note that \(\tau\) is sometimes used to refer to \(2 \times \pi\) but that usage has nothing to do with this post.