Jordan–Pólya number

 Every Jordan–Pólya number , except 2, has the property that its factorial  can be written as a product of smaller factorials. 

This can be done simply by expanding  and then replacing  in this product by its representation as a product of factorials.

 It is conjectured, but unproven, that the only numbers  whose factorial  equals a product of smaller factorials are the Jordan–Pólya numbers (except 2) and the two exceptional numbers 9 and 10,

for which  

and . 

The only other known representation of a factorial as a product of smaller factorials, not obtained by replacing  in the product expansion of , is 

, 

but as  is itself a Jordan–Pólya number, it also has the representation 

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, 40320, 41472, 46080, 46656, 49152, 51840, 55296, 57600, 60480, 61440, 62208, 65536, 69120, 73728, 80640, 82944, 86400, 92160, 93312, 98304, 103680, 110592, 115200, 120960, 122880, 124416, 131072, 138240, 147456, 155520, 161280, 165888, 172800, 181440, 184320, 186624, 196608, 207360, 221184, 230400, 241920, 245760, 248832, 262144, 276480, 279936, 294912, 311040, 322560, 331776, 345600, 362880, 368640, 373248, 393216, 414720, 442368, 460800, 483840, 491520, 497664, 518400, 524288, 552960, 559872, 589824, 604800, 622080, 645120, 663552, 691200, 725760, 737280, 746496, 786432, 829440, 884736, 921600, 933120, 967680, 983040, 995328, 1036800, 1048576, 1088640, 1105920, 1119744, 1179648, 

Liste [nombre-factorielle, exposant de la puissance de 2, la puissance de 2]

[2, 1, 2], [3, 1, 2], [4, 3, 8], [5, 3, 8], [6, 4, 16], [7, 4, 16], [8, 7, 128], [9, 7, 128], [10, 8, 256], [11, 8, 256], [12, 10, 1024], [13, 10, 1024], [14, 11, 2048], [15, 11, 2048], [16, 15, 32768], [17, 15, 32768], [18, 16, 65536], [19, 16, 65536], [20, 18, 262144], …

w

13!=1024*768=2^10×3^5×5^2×7×11×13=6227020800

768=2^8*3