Highly totient number

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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.[1]

The totient of a number x, with prime factorization x=\prod_i p_i^{e_i}, is the product

\phi(x)=\prod_i (p_i-1)p_i^{e_i-1}.

Thus, a highly totient number is a number that has more ways of being expressed as a product of this form than does any smaller number.

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.

Another definition of "highly totient number" is a number n such that φ(m)<φ(n) when m<n, i.e. its totient is greater than the totient of any smaller number. These numbers are conjectured to be just the primes.

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