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], … |
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