Jordan-Polya Numbers

The number associated with my diurnal age today, 27648, is a Jordan-Polya number which Numbers Aplenty defines as a number that can be written as the product of factorial numbers. In the case of 27648 we have:$$\begin{align} 27648 &=2^{10} \times 3^3\\&=(4!)^3 \times 2! \end{align} $$There aren't that many of them in the range up to 40,000. Here are the initial numbers:

1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, 72, 96, 120, 128, 144, 192, 216, 240, 256, 288, 384, 432, 480, 512, 576, 720, 768, 864, 960, 1024, 1152, 1296, 1440, 1536, 1728, 1920, 2048, 2304, 2592, 2880, 3072, 3456, 3840, 4096, 4320, 4608, 5040, 5184, 5760, 6144, 6912, 7680, 7776, 8192, 8640, 9216, 10080, 10368, 11520, 12288, 13824, 14400, 15360, 15552, 16384, 17280, 18432, 20160, 20736, 23040, 24576, 25920, 27648, 28800, 30240, 30720, 31104, 32768, 34560, 36864

Here is what Wikipedia had to say about them:

In mathematics, the Jordan–Pólya numbers are the numbers that can be obtained by multiplying together one or more factorials, not required to be distinct from each other. For instance, 480 is a Jordan–Pólya number because:$$480 =2! \times 2! \times 5!$$Every tree has a number of symmetries that is a Jordan–Pólya number, and every Jordan–Pólya number arises in this way as the order of an automorphism group of a tree. These numbers are named after Camille Jordan and George Pólya, who both wrote about them in the context of symmetries of trees.

These numbers grow more quickly than polynomials but more slowly than exponentials. As well as in the symmetries of trees, they arise as the numbers of transitive orientations of comparability graphs and in the problem of finding factorials that can be represented as products of smaller factorials.

A tree that (as an abstract graph) has 480 symmetries (automorphisms). There are 2 ways of permuting the two children of the upper left vertex, 2 ways of permuting the two children of the upper middle vertex, and 5! = 120 ways of permuting the five children of the upper right vertex, for 2 x 2 x 120 = 480 symmetries altogether.

George Pólya
13 December 1887 - 7 September 1985

George Pólya was one of the greatest mathematicians of the 20th century and there is a brief biography of his life to be found here. To quote from this article:

With no hesitation, George Pólya is my personal hero as a mathematician. ... [he] is not only a distinguished gentleman but a most kind and gentle man: his ebullient enthusiasm, the twinkle in his eye, his tremendous curiosity, his generosity with his time, his spry energetic walk, his warm genuine friendliness, his welcoming visitors into his home and showing them his pictures of great mathematicians he has known - these are all components of his happy personality. As a mathematician, his depth, speed, brilliance, versatility, power and universality are all inspiring. Would that there were a way of teaching and learning these traits.

Camille Jordan
5th January 1838 - 22nd January 1922

Camille Jordan made most of his mathematical contributions in the 19th century and a brief biography can be found here. It can be noted that, quoting from the article:

The Journal de Mathématiques Pure et Appliquées was a leading mathematical journal and played a very significant part in the development of mathematics throughout the 19th century. It was usually known as the Journal de Liouville since Liouville had founded the journal in 1836. Liouville died in 1882 and in 1885 Jordan became editor of the Journal, a role he kept for over 35 years until his death.