Highly cototient number

In number theory, a branch of mathematics, a highly cototient number is a positive integer "k" which is above one and has more solutions to the equation

:"x" − φ("x") = "k",

than any other integer below "k" and above one. Here, φ is Euler's totient function. There are infinitely many solutions to the equation for "k" = 1 so this value is excluded in the definition. The first few highly cototient numbers are:

:2, 4, 8, 23, 35, 47, 59, 63, 83, 89, 113, 119, 167, 209, 269, 299, 329, 389, 419, 509, 629, 659, 779, 839, 1049, 1169, 1259, 1469, 1649, 1679, 1889 OEIS|id=A100827.

There are many odd highly cototient numbers. In fact, after 8, all the numbers listed above are odd, and after 167 all the numbers listed above are congruent to 9 mod 10.

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 does, as the numbers get larger.

The first few highly cototient numbers which are primes OEIS|id=A105440 are

:2, 23, 47, 59, 83, 89, 113, 167, 269, 389, 419, 509, 659, 839.