A number which can be written as a product of factorials. more
The first 600 Jordan-Polya 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,
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,
1209600,
1244160,
1290240,
1327104,
1382400,
1451520,
1474560,
1492992,
1572864,
1658880,
1679616,
1728000,
1769472,
1843200,
1866240,
1935360,
1966080,
1990656,
2073600,
2097152,
2177280,
2211840,
2239488,
2359296,
2419200,
2488320,
2580480,
2654208,
2764800,
2903040,
2949120,
2985984,
3110400,
3145728,
3317760,
3359232,
3456000,
3538944,
3628800,
3686400,
3732480,
3870720,
3932160,
3981312,
4147200,
4194304,
4354560,
4423680,
4478976,
4718592,
4838400,
4976640,
5160960,
5308416,
5529600,
5598720,
5806080,
5898240,
5971968,
6220800,
6291456,
6531840,
6635520,
6718464,
6912000,
7077888,
7257600,
7372800,
7464960,
7741440,
7864320,
7962624,
8294400,
8388608,
8709120,
8847360,
8957952,
9437184,
9676800,
9953280,
10077696,
10321920,
10368000,
10616832,
11059200,
11197440,
11612160,
11796480,
11943936,
12441600,
12582912,
13063680,
13271040,
13436928,
13824000,
14155776,
14515200,
14745600,
14929920,
15482880,
15728640,
15925248,
16588800,
16777216,
17418240,
17694720,
17915904,
18662400,
18874368,
19353600,
19906560,
20155392,
20643840,
20736000,
21233664,
21772800,
22118400,
22394880,
23224320,
23592960,
23887872,
24883200,
25165824,
25401600,
26127360,
26542080,
26873856,
27648000,
28311552,
29030400,
29491200,
29859840,
30965760,
31457280,
31850496,
33177600,
33554432,
33592320,
34836480,
35389440,
35831808,
37324800,
37748736,
38707200,
39191040,
39813120,
39916800,
40310784,
41287680,
41472000,
42467328,
43545600,
44236800,
44789760,
46448640,
47185920,
47775744,
49766400,
50331648,
50803200,
52254720,
53084160,
53747712,
55296000,
56623104,
58060800,
58982400,
59719680,
60466176,
61931520,
62208000,
62914560,
63700992,
66355200,
67108864,
67184640,
69672960,
70778880,
71663616,
72576000,
74649600,
75497472,
77414400,
78382080,
79626240,
79833600,
80621568,
82575360,
82944000,
84934656,
87091200,
88473600,
89579520,
92897280,
94371840,
95551488,
99532800,
100663296,
101606400,
104509440,
106168320,
107495424,
110592000,
111974400,
113246208,
116121600,
117964800,
119439360,
120932352,
123863040,
124416000,
125829120,
127401984,
130636800,
132710400,
134217728,
134369280,
139345920,
141557760,
143327232,
145152000,
149299200,
150994944,
152409600,
154828800,
156764160,
159252480,
159667200,
161243136,
165150720,
165888000,
169869312,
174182400,
176947200,
179159040,
185794560,
188743680,
191102976,
199065600,
201326592,
201553920,
203212800,
207360000,
209018880,
212336640,
214990848,
221184000,
223948800,
226492416,
232243200,
235146240,
235929600,
238878720,
239500800,
241864704,
247726080,
248832000,
251658240,
254803968,
261273600,
265420800,
268435456,
268738560,
278691840,
283115520,
286654464,
290304000,
298598400,
301989888,
304819200,
309657600,
313528320,
318504960,
319334400,
322486272,
330301440,
331776000,
339738624,
348364800,
353894400,
358318080,
362797056,
371589120,
373248000,
377487360,
382205952,
398131200,
402653184,
403107840,
406425600,
414720000,
418037760,
424673280,
429981696,
435456000,
442368000,
447897600,
452984832,
464486400,
470292480,
471859200,
477757440,
479001600,
483729408,
495452160,
497664000,
503316480,
509607936,
522547200,
530841600,
536870912,
537477120,
557383680,
566231040,
573308928,
580608000,
597196800,
603979776,
609638400,
619315200,
627056640,
637009920,
638668800,
644972544,
660602880,
663552000,
671846400,
679477248,
696729600,
707788800,
716636160,
725594112,
743178240,
746496000,
754974720,
764411904,
783820800,
796262400,
805306368,
806215680,
812851200,
829440000,
836075520,
849346560,
859963392,
870912000,
884736000,
895795200,
905969664,
914457600,
928972800,
940584960,
943718400,
955514880,
958003200,
967458816,
990904320,
995328000,
1006632960,
1019215872,
1045094400,
1061683200,
1073741824,
1074954240,
1114767360,
1132462080,
1146617856,
1161216000,
1194393600,
1207959552,
1209323520,
1219276800,
1238630400,
1244160000,
1254113280,
1274019840,
1277337600,
1289945088,
1321205760,
1327104000,
1343692800,
1358954496,
1393459200,
1410877440,
1415577600,
1433272320,
1437004800,
1451188224,
1486356480,
1492992000,
1509949440,
1528823808,
1567641600,
1592524800,
1610612736,
1612431360,
1625702400,
1658880000,
1672151040,
1698693120,
1719926784,
1741824000,
1769472000,
1791590400,
1811939328,
1828915200,
1857945600,
1881169920,
1887436800,
1911029760,
1916006400,
1934917632,
1981808640,
1990656000,
2013265920,
2038431744,
2090188800,
2123366400,
2147483648,
2149908480,
2176782336,
2229534720,
2239488000,
2264924160,
2293235712,
2322432000,
2388787200,
2415919104,
2418647040,
2438553600.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 3065 values, from 1 to 995515121664000).
n\r | 0 | 1 |
2 | 3064 | 1 | 2 |
3 | 3015 | 25 | 25 | 3 |
4 | 3062 | 1 | 2 | 0 | 4 |
5 | 2550 | 136 | 131 | 122 | 126 | 5 |
6 | 3015 | 1 | 25 | 0 | 24 | 0 | 6 |
7 | 1513 | 291 | 277 | 229 | 262 | 240 | 253 | 7 |
8 | 3059 | 1 | 1 | 0 | 3 | 0 | 1 | 0 | 8 |
9 | 2924 | 9 | 9 | 46 | 8 | 8 | 45 | 8 | 8 | 9 |
10 | 2550 | 1 | 131 | 0 | 126 | 0 | 135 | 0 | 122 | 0 | 10 |
11 | 473 | 261 | 261 | 258 | 261 | 259 | 258 | 258 | 258 | 259 | 259 |
A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.