In number theory, a branch of mathematics, a highly cototient number k is an integer that has more solutions to the equation x - φ(x) = k, where φ is Euler's totient function, than any integer below it, with the exception of 1. In other words, k is a cototient more often than any integer below it except 1. 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 (sequence A100827 in OEIS)

1 is the highest cototient number, a k with infinitely many more solutions to x - φ(x) = k than any other integer. While 1 is the smallest odd highly totient number, it is not the only odd highly cototient number. 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, though the highly cototient numbers get tougher to find the higher one goes, since calculating the totient function involves factorization into primes, something that becomes extremely difficult as the numbers get larger.

Integer sequences