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.
Famous quotes containing the words highly and/or number:
“The truth is, as every one knows, that the great artists of the world are never Puritans, and seldom even ordinarily respectable. No virtuous man—that is, virtuous in the Y.M.C.A. sense—has ever painted a picture worth looking at, or written a symphony worth hearing, or a book worth reading, and it is highly improbable that the thing has ever been done by a virtuous woman.”
—H.L. (Henry Lewis)
“In this world, which is so plainly the antechamber of another, there are no happy men. The true division of humanity is between those who live in light and those who live in darkness. Our aim must be to diminish the number of the latter and increase the number of the former. That is why we demand education and knowledge.”
—Victor Hugo (1802–1885)