The highly composite numbers up to 1015 :
1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160, 2882880, 3603600, 4324320, 6486480, 7207200, 8648640, 10810800, 14414400, 17297280, 21621600, 32432400, 36756720, 43243200, 61261200, 73513440, 110270160, 122522400, 147026880, 183783600, 245044800, 294053760, 367567200, 551350800, 698377680, 735134400, 1102701600, 1396755360, 2095133040, 2205403200, 2327925600, 2793510720, 3491888400, 4655851200, 5587021440, 6983776800, 10475665200, 13967553600, 20951330400, 27935107200, 41902660800, 48886437600, 64250746560, 73329656400, 80313433200, 97772875200, 128501493120, 146659312800, 160626866400, 240940299600, 293318625600, 321253732800, 481880599200, 642507465600, 963761198400, 1124388064800, 1606268664000, 1686582097200, 1927522396800, 2248776129600, 3212537328000, 3373164194400, 4497552259200, 6746328388800, 8995104518400, 9316358251200, 13492656777600, 18632716502400, 26985313555200, 27949074753600, 32607253879200, 46581791256000, 48910880818800, 55898149507200, 65214507758400, 93163582512000, 97821761637600, 130429015516800, 195643523275200, 260858031033600, 288807105787200, 391287046550400, 577614211574400, 782574093100800, 866421317361600.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 125 values, from 1 to 866421317361600).
| n\r | 0 | 1 | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 124 | 1 | 2 | ||||||||
| 3 | 122 | 2 | 1 | 3 | |||||||
| 4 | 122 | 1 | 2 | 0 | 4 | ||||||
| 5 | 117 | 3 | 2 | 1 | 2 | 5 | |||||
| 6 | 122 | 1 | 1 | 0 | 1 | 0 | 6 | ||||
| 7 | 111 | 3 | 2 | 2 | 2 | 2 | 3 | 7 | |||
| 8 | 116 | 1 | 1 | 0 | 6 | 0 | 1 | 0 | 8 | ||
| 9 | 113 | 1 | 1 | 4 | 1 | 0 | 5 | 0 | 0 | 9 | |
| 10 | 117 | 1 | 2 | 0 | 2 | 0 | 2 | 0 | 1 | 0 | 10 |
| 11 | 99 | 3 | 3 | 2 | 5 | 2 | 3 | 1 | 3 | 2 | 2 |
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.
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