The trimorphic numbers up to 1015 :
1, 4, 5, 6, 9, 24, 25, 49, 51, 75, 76, 99, 125, 249, 251, 375, 376, 499, 501, 624, 625, 749, 751, 875, 999, 1249, 3751, 4375, 4999, 5001, 5625, 6249, 8751, 9375, 9376, 9999, 18751, 31249, 40625, 49999, 50001, 59375, 68751, 81249, 90624, 90625, 99999, 109375, 109376, 218751, 281249, 390625, 499999, 500001, 609375, 718751, 781249, 890624, 890625, 999999, 2109375, 2890624, 2890625, 4218751, 4999999, 5000001, 5781249, 7109375, 7109376, 7890625, 9218751, 9999999, 12890624, 12890625, 24218751, 25781249, 37109375, 49999999, 50000001, 62890625, 74218751, 75781249, 87109375, 87109376, 99999999, 212890624, 212890625, 287109375, 425781249, 499999999, 500000001, 574218751, 712890625, 787109375, 787109376, 925781249, 999999999, 1425781249, 1787109375, 1787109376, 3212890625, 3574218751, 4999999999, 5000000001, 6425781249, 6787109375, 8212890624, 8212890625, 8574218751, 9999999999, 13574218751, 18212890624, 18212890625, 31787109375, 36425781249, 49999999999, 50000000001, 63574218751, 68212890625, 81787109375, 81787109376, 86425781249, 99999999999, 163574218751, 336425781249, 418212890625, 499999999999, 500000000001, 581787109375, 663574218751, 836425781249, 918212890624, 918212890625, 999999999999, 4836425781249, 4918212890625, 4999999999999, 5000000000001, 5081787109375, 5163574218751, 9836425781249, 9918212890624, 9918212890625, 9999999999999, 19836425781249, 30163574218751, 40081787109375, 40081787109376, 49999999999999, 50000000000001, 59918212890624, 59918212890625, 69836425781249, 80163574218751, 90081787109375, 99999999999999, 240081787109375, 259918212890624, 259918212890625, 480163574218751, 499999999999999, 500000000000001, 519836425781249, 740081787109375, 740081787109376, 759918212890625, 980163574218751, 999999999999999.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 168 values, from 1 to 999999999999999).
| n\r | 0 | 1 | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 26 | 142 | 2 | ||||||||
| 3 | 69 | 57 | 42 | 3 | |||||||
| 4 | 25 | 66 | 1 | 76 | 4 | ||||||
| 5 | 50 | 51 | 0 | 0 | 67 | 5 | |||||
| 6 | 9 | 46 | 6 | 60 | 11 | 36 | 6 | ||||
| 7 | 22 | 24 | 24 | 23 | 30 | 29 | 16 | 7 | |||
| 8 | 23 | 62 | 0 | 6 | 2 | 4 | 1 | 70 | 8 | ||
| 9 | 21 | 7 | 16 | 16 | 32 | 10 | 32 | 18 | 16 | 9 | |
| 10 | 0 | 39 | 0 | 0 | 14 | 50 | 12 | 0 | 0 | 53 | 10 |
| 11 | 16 | 15 | 12 | 11 | 17 | 18 | 20 | 18 | 12 | 18 | 11 |
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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