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:^[1]

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 the OEIS)

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 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 does, as the numbers get larger.

Example

The cototient of x is defined as x – φ(x), i.e. the number of positive integers less than or equal to x that have at least one prime factor in common with x. For example, the cototient of 6 is 4 since these 4 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 (one number in each case), so 4 is a highly cototient number.

(sequence A063740 in the OEIS)

Primes

The first few highly cototient numbers which are primes are ^[2]

2, 23, 47, 59, 83, 89, 113, 167, 269, 389, 419, 509, 659, 839, 1049, 1259, 1889, 2099, 2309, 2729, 3359, 3989, 4289, 4409, 5879, 6089, 6719, 9029, 9239, ... (sequence A105440 in the OEIS)

References