Highly cototient number
In number theory, a branch of mathematics, a highly cototient number is a positive integer which is above 1 and has more solutions to the equation
than any other integer below and above 1. Here, is Euler's totient function. There are infinitely many solutions to the equation for
so this value is excluded in the definition. The first few highly cototient numbers are:
Many of the highly cototient numbers are odd. In fact, after 8, all the numbers listed above are odd, and after 167 all the numbers listed above are congruent to 29 modulo 30.
The concept is somewhat analogous to that of highly composite numbers. Just as there are infinitely many highly composite numbers, there are also infinitely many highly cototient numbers. Computations become harder, since integer factorization becomes harder as the numbers get larger.
Example
The cototient of is defined as, i.e. the number of positive integers less than or equal to that have at least one prime factor in common with. For example, the cototient of 6 is 4 since these four positive integers have a prime factor in common with 6: 2, 3, 4, 6. The cototient of 8 is also 4, this time with these integers: 2, 4, 6, 8. There are exactly two numbers, 6 and 8, which have cototient 4. There are fewer numbers which have cototient 2 and cototient 3, so 4 is a highly cototient number.| n | ks such that | number of ks such that |
| 0 | 1 | 1 |
| 1 | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,... | ∞ |
| 2 | 4 | 1 |
| 3 | 9 | 1 |
| 4 | 6, 8 | 2 |
| 5 | 25 | 1 |
| 6 | 10 | 1 |
| 7 | 15, 49 | 2 |
| 8 | 12, 14, 16 | 3 |
| 9 | 21, 27 | 2 |
| 10 | 0 | |
| 11 | 35, 121 | 2 |
| 12 | 18, 20, 22 | 3 |
| 13 | 33, 169 | 2 |
| 14 | 26 | 1 |
| 15 | 39, 55 | 2 |
| 16 | 24, 28, 32 | 3 |
| 17 | 65, 77, 289 | 3 |
| 18 | 34 | 1 |
| 19 | 51, 91, 361 | 3 |
| 20 | 38 | 1 |
| 21 | 45, 57, 85 | 3 |
| 22 | 30 | 1 |
| 23 | 95, 119, 143, 529 | 4 |
| 24 | 36, 40, 44, 46 | 4 |
| 25 | 69, 125, 133 | 3 |
| 26 | 0 | |
| 27 | 63, 81, 115, 187 | 4 |
| 28 | 52 | 1 |
| 29 | 161, 209, 221, 841 | 4 |
| 30 | 42, 50, 58 | 3 |
| 31 | 87, 247, 961 | 3 |
| 32 | 48, 56, 62, 64 | 4 |
| 33 | 93, 145, 253 | 3 |
| 34 | 0 | |
| 35 | 75, 155, 203, 299, 323 | 5 |
| 36 | 54, 68 | 2 |
| 37 | 217, 1369 | 2 |
| 38 | 74 | 1 |
| 39 | 99, 111, 319, 391 | 4 |
| 40 | 76 | 1 |
| 41 | 185, 341, 377, 437, 1681 | 5 |
| 42 | 82 | 1 |
| 43 | 123, 259, 403, 1849 | 4 |
| 44 | 60, 86 | 2 |
| 45 | 117, 129, 205, 493 | 4 |
| 46 | 66, 70 | 2 |
| 47 | 215, 287, 407, 527, 551, 2209 | 6 |
| 48 | 72, 80, 88, 92, 94 | 5 |
| 49 | 141, 301, 343, 481, 589 | 5 |
| 50 | 0 |