Ahighly totient number k is an integer that has more solutions to the equation φ(x) = k, where φisEuler'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
with 1, 3, 4, 5, 6, 10, 1, 17, 21, 31, 34, 37, 38, 49, 54, and 72 totient solutions respectively.
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.
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