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:
“Imagine believing in the control of inflation by curbing the money supply! That is like deciding to stop your dog fouling the sidewalk by plugging up its rear end. It is highly unlikely to succeed, but if it does it kills the hound.”
—Michael D. Stephens. “On Sinai, There’s No Economics,” New York Times (Nov. 13, 1981)
“The best number for a dinner party is two—myself and a dam’ good head waiter.”
—Nubar Gulbenkian (1896–1972)