Jordan-Polya numbers

A number

is a Jordan-Polya number if it can be written as the product of factorial numbers.

For example, $92160$ is a Jordan-Polya number, because it can be written as $(2!)^7\cdot 6!$ .

Jordan-Polya numbers arise in the following simple combinatorial problem. If $k$ groups of $n_1,n_2,\dots,n_k$ distinct objects are given, then the number of distinct permutations of the $n_1+n_2+\cdots+n_k$ objects which maintain objects of the same group adjacent are $k!\cdot n_1!\cdot n_2!\cdots n_k!$ , a Jordan-Polya number.

The first Jordan-Polya numbers are 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 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many Jordan-Polya numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Factorial Products
OEIS, Sequence A001013

Jordan-Polya numbers can also be... (you may click on names or numbers and on + to get more values)

ABA 24 32 64 72 + 9906778275840 aban 12 16 24 32 + 960 abundant 12 24 36 48 + 49766400 Achilles 72 288 432 864 + 49941577728000 admirable 12 24 120 30240 alternating 12 16 32 36 + 147456 amenable 12 16 24 32 + 995328000 apocalyptic 192 384 720 864 + 28800 arithmetic 96 432 480 864 + 9953280 astonishing 216 betrothed 48 binomial 36 120 c.pentagonal 16 c.triangular 64 cake 64 576 2048 canyon 216 512 768 1024 + 98304 compositorial 24 192 1728 17280 + 115880067072000 congruent 24 96 120 216 + 9953280 constructible 12 16 24 32 + 844424930131968 cube 64 216 512 1728 + 949978046398464 Cunningham 24 48 120 288 + 25920 d-powerful 24 2048 24576 345600 7864320 deficient 16 32 64 128 + 8388608 dig.balanced 12 120 216 240 + 8847360 double fact. 48 384 3840 46080 + 51011754393600 droll 72 240 5184 17280 + 847579919155200 Duffinian 16 32 36 64 + 9437184 eban 32 36 64 economical 16 32 64 128 + 19906560 enlightened 256 2048 2304 262144 + 274877906944 equidigital 16 32 64 192 + 19353600 eRAP 24 esthetic 12 32 432 3456 Eulerian 120 evil 12 24 36 48 + 995328000 factorial 24 120 720 5040 + 355687428096000 Fibonacci 144 Friedman 128 216 1024 1296 + 995328 frugal 128 256 512 1024 + 995328000 gapful 120 192 240 480 + 99632332800 happy 32 192 1152 1920 + 9437184 harmonic 30240 Harshad 12 24 36 48 + 9953280000 hexagonal 120 highly composite 12 24 36 48 + 20160 house 32 hungry 144 iban 12 24 72 120 + 414720 idoneal 12 16 24 48 + 240 impolite 16 32 64 128 + 562949953421312 inconsummate 216 432 1536 2592 + 829440 interprime 12 64 72 120 + 74649600 junction 216 1024 1920 13824 + 94371840 katadrome 32 64 72 96 + 8640 Leyland 32 512 93312 33554432 17832200896512 lonely 120 Lynch-Bell 12 24 36 48 + 18432 magnanimous 12 16 32 512 metadrome 12 16 24 36 + 3456 mountain 120 192 240 384 + 7864320 nialpdrome 32 64 72 96 + 86400 nonagonal 24 nude 12 24 36 48 + 429981696 O'Halloran 12 36 oban 12 16 36 96 + 960 octagonal 96 8640 odious 16 32 64 128 + 967458816 pancake 16 4096 panconsummate 12 24 36 72 144 pandigital 120 216 30720 pentagonal 12 pernicious 12 24 36 48 + 9676800 Perrin 12 plaindrome 12 16 24 36 + 12288 power 16 32 36 64 + 47775744000000 powerful 16 32 36 64 + 995515121664000 practical 12 16 24 32 + 9953280 prim.abundant 12 productive 12 16 36 256 pronic 12 72 240 pseudoperfect 12 24 36 48 + 995328 Ruth-Aaron 16 24 Saint-Exupery 480 3840 30720 103680 + 974098582732800 self 64 288 512 1728 + 905969664 Smith 576 7077888 8957952 square 16 36 64 144 + 989560464998400 straight-line 432 864 3456 strobogrammatic 96 super Niven 12 24 36 48 + 604800 super-d 13824 16384 138240 161280 + 4838400 superabundant 12 24 36 48 + 10080 tau 12 24 36 72 + 995328000 tetrahedral 120 triangular 36 120 tribonacci 24 trimorphic 24 uban 12 16 32 36 + 96 Ulam 16 36 48 72 + 8294400 unprimeable 512 1920 2048 5040 + 8847360 untouchable 96 120 216 288 + 967680 upside-down 64 8192 vampire 186624 wasteful 12 24 36 48 + 9676800 Zuckerman 12 24 36 128 + 429981696 Zumkeller 12 24 48 96 + 98304