Strongly Refactorable Numbers

On November 21st 2022, I posted about Tau Numbers and wrote:

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 \( \tau(n) | n \). 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. Source.

So a refactorable number is the same as a tau number and these are relatively common. Up to 100,000, there are 5257 such numbers representing 5.257% of the range. However, they have a natural density of zero. So what is a strongly refactorable number?

I discovered what characterised these numbers thanks to a property of the number associated with my diurnal age today, 27360, that also corresponds to the 29th of February 2024. It happens to be a member of OEIS A141586:

A141586

Strongly refactorable numbers: numbers \(n\) such that if \(n\) is divisible by \(d\), it is divisible by the number of divisors of \(d\).

Such numbers are few and far between. The number previous to 27360 was 25440. The 41 members of the sequence up to 40,000 are (permalink):

1, 2, 12, 24, 36, 72, 240, 480, 720, 1440, 3360, 4320, 5280, 6240, 6720, 8160, 9120, 10080, 11040, 13440, 13920, 14880, 15840, 17760, 18720, 19680, 20160, 20640, 21600, 22560, 24480, 25440, 27360, 28320, 29280, 32160, 33120, 34080, 35040, 37920, 39840

27360 factorises to \(2^5 \times 3^2 \times 5 \times 19 \) and thus it has 72 divisors. These are:

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 19, 20, 24, 30, 32, 36, 38, 40, 45, 48, 57, 60, 72, 76, 80, 90, 95, 96, 114, 120, 144, 152, 160, 171, 180, 190, 228, 240, 285, 288, 304, 342, 360, 380, 456, 480, 570, 608, 684, 720, 760, 855, 912, 1140, 1368, 1440, 1520, 1710, 1824, 2280, 2736, 3040, 3420, 4560, 5472, 6840, 9120, 13680, 27360

The number of divisors of these divisors is listed below and all these numbers divide 27360 as well:

1, 2, 2, 3, 2, 4, 4, 3, 4, 6, 4, 5, 6, 2, 6, 8, 8, 6, 9, 4, 8, 6, 10, 4, 12, 12, 6, 10, 12, 4, 12, 8, 16, 15, 8, 12, 6, 18, 8, 12, 20, 8, 18, 10, 12, 24, 12, 16, 24, 16, 12, 18, 30, 16, 12, 20, 24, 24, 36, 20, 24, 24, 32, 30, 24, 36, 40, 36, 48, 48, 60, 72

It will be 960 days before I see the next strongly refactorable number (28320). That will occur on October 16th, 2026.