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Lucky Numbers
© Walter Schneider 2001
(last updated 24/12/2002)
Lucky numbers are defined by a variation of the well-known sieve of Eratosthenes. Beginning with the natural numbers strike out all even ones, leaving the odd numbers 1, 3, 5, 7, 9, 11, 13, ... The second number is 3, next strike out every third number, leaving 1, 3, 7, 9, 13, ... The third number is 7, next strike out every seventh number a.s.o. The numbers surviving are called lucky numbers. The first lucky numbers are (Sloane's A000959):
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, 105, 111, 115, 127, 129, 133, 135, 141, 151, 159, 163, 169, 171, 189, 193, 195, 201, 205, 211, 219, 223, 231, 235, 237, 241, 259, 261, 267, 273, 283, 285, 289, 297, 303, 307, 319, 321, 327, 331, 339, 349, 357, 361, 367, 385, 391, 393, 399, ...
A list of the first 1000 lucky numbers is available here.
What's most interesting about lucky numbers is the fact that they share a lot of properties with primes. As can be seen from the next table the density of the lucky numbers is close to the density of the primes. This seems also be true for the density of the twin luckies and the twin primes. In addition a lot of conjectures about primes seem also to be true for the luckies. For example one of the most famous ones, the Goldbach conjecture, stating that each even integer is the sum of at most two primes seems also to be true.
Because of the many similarities between primes and luckies it seems that a lot of the properties of the primes are just a result of the sieving process!
In November 2001 I calculated all luckies up to 109 and verified the Goldbach conjecture up to this limit. Note that the sieving process is more complicated for the luckies because for primes only a sieving up to the square root has to be done. In May 2002 the search was extended to 1010 again verifying the Goldbach conjecture. The largest lucky found is 9,999,999,997 and this should be the largest known lucky number.
| The Number of Luckies Compared to Primes (and the Largest Luckies up to the Search Limit) |
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|---|---|---|---|---|---|
| Limit | Number of Luckies |
Number of Primes |
Number of Twin Luckies |
Number of Twin Primes |
Largest Lucky |
| 103 | 153 | 168 | 33 | 35 | 997 |
| 104 | 1,118 | 1,229 | 178 | 205 | 9,999 |
| 105 | 8,772 | 9,592 | 1,162 | 1,224 | 99,987 |
| 106 | 71,918 | 78,498 | 7,669 | 8,169 | 999,987 |
| 107 | 609,237 | 664,579 | 55,548 | 58,980 | 9,999,997 |
| 108 | 5,286,238 | 5,761,455 | 419,174 | 440,312 | 99,999,979 |
| 109 | 46,697,909 | 50,847,534 | 3,274,570 | 3,424,506 | 999,999,991 |
| 1010 | 418,348,044 | 455,052,511 | 26,298,112 | 27,412,679 | 9,999,999,997 |
Note that the first two steps in the sieving process eliminate all numbers of the form 3k + 2. Therefore no lucky number can have a digital root of 2, 5 and 8. This fact can sometimes be used to determine quickly that a given number is not lucky.
