Highly totient number

A highly totient number k is an integer that has more solutions to the equation φ(x) = k, where φ is Euler's totient function, than any integer below it. The first few highly totient numbers are

1, 2, 4, 8, 12, 24, 48, 72, 144, 240, 432, 480, 576, 720, 1152, 1440 (sequence A097942 in OEIS).

with 1, 3, 4, 5, 6, 10, 11, 17, 21, 31, 34, 37, 38, 49, 54, and 72 totient solutions respectively. The sequence of highly totient numbers is a subset of the sequence of smallest number k with exactly n solutions to φ(x) = k.

These numbers have more ways of being expressed as products of numbers of the form p - 1 and their products than smaller integers.

The concept is somewhat analogous to that of highly composite numbers, and in the same way that 1 is the only odd highly composite number, it is also the only odd highly totient number (indeed, the only odd number to not be a nontotient). And just as there are infinitely many highly composite numbers, there are also infinitely many highly totient numbers, though the highly totient 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.
See also

Highly composite number

References

L. Havelock, A Few Observations on Totient and Cototient Valence from PlanetMath