Anti-tau Numbers

I was familiar with what a tau number was, the definition being that it is an integer divisible by its number of its divisors.  Tau is the divisor function and returns the number of divisors of an integer e.g. \( \tau(6)=4\) because 6 has four divisors (1, 2, 3 and 6). However, 6 is not a tau number because 4 does not divide 6 without remainder.

In general, we can say that a number \(n\) is a tau number if \( \tau(n)|n\). 12 is a tau number because \( \tau(12) =6\) and \(6|12=2\). An alternative term, refactorable number, can be used for tau numbers (see Wikipedia entry). What then is an anti-tau number? Well, it turns out to be a number \(n\) such that \( \text{gcd}(n,\tau(n))=1\). These numbers comprise OEIS A046642.

A046642

Numbers \(k\) such that \(k\) and number of divisors \( \text{d}(k)\) are relatively prime.

Note that \( \text{d}(k)\) is sometimes used in place of \( \tau(k) \). All odd prime numbers will be anti-tau numbers because they have two divisors and 2 does not divide them. An example of an anti-tau number is 15 because it has four divisors (1, 3, 5 and 15) and gcd(15, 4) = 1. These numbers are quite frequent. There are 18080 in the range up to 40,000, representing over 45% of all the numbers in the range. The initial members are:

1, 3, 4, 5, 7, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 100, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131

Notice that all even anti-tau numbers are square numbers (4, 16, 64, 100, 196, 256, 484, 676, 784, 1024 etc.) of the form \(4k^2\) where \(k\) is a positive even integer. However, not all even values of \(k\) produce anti-tau numbers. The initial values of \(k\) are 2, 4, 8, 10, 14, 16,  22, ...

By contrast, tau numbers are less common. In the range up to 100,000, there are 5257 of them representing 5.257% of the range. The anti-tau numbers were brought to my attention by one of the properties of the number representing my diurnal age, 26895, qualifying it for membership of OEIS A341780:

A341780

Starts of runs of 3 consecutive anti-tau numbers (A046642).

Let's look at 26895, 26896 and 26897 to confirm that this is true:

• 26895 = 3 x 5 x 11 x 163 with 16 divisors and 16 does not divide the number
• 26896 = 2^4 x 41^2 with 15 divisors and 15 does not divide the number
• 26897 = 13 x 2069 with 4 divisors and 4 does not divide the number

Despite how common anti-tau numbers are, these runs of three (the maximum possible) are not that frequent. The problem is that the runs are of the form odd-even-odd and the even numbers must be of the form \(4k^2\) as mentioned earlier. It is these relatively sparse even anti-tau numbers that make these runs less common than might be expected. Notice how \(26896 = 4 \times 82^2\). There are only 69 such numbers in the range up to 100,000. Here is the list:

3, 15, 195, 255, 483, 783, 1023, 1155, 1295, 1443, 1599, 2703, 3363, 4623, 4899, 5183, 6399, 6723, 7395, 7743, 8463, 8835, 10815, 11235, 11663, 12099, 12543, 15375, 16383, 16899, 17955, 18495, 20163, 24963, 25599, 26895, 27555, 31683, 33855, 35343, 36099, 37635, 38415, 40803, 44943, 45795, 46655, 47523, 52899, 53823, 55695, 56643, 61503, 62499, 64515, 65535, 70755, 71823, 73983, 80655, 81795, 82943, 85263, 87615, 88803, 91203, 92415, 94863, 96099

Here is a permalink that will confirm all the above results.