In number theory, a branch of mathematics, a highly cototient number is a positive integer k which is above one and has more solutions to the equation
- x − φ(x) = k,
than any other integer below k and above one. Here, φ is Euler's totient function. There are infinitely many solutions to the equation for k = 1 so this value is excluded in the definition. The first few highly cototient numbers are:
- 2, 4, 8, 23, 35, 47, 59, 63, 83, 89, 113, 119, 167, 209, 269, 299, 329, 389, 419, 509, 629, 659, 779, 839, 1049, 1169, 1259, 1469, 1649, 1679, 1889 (sequence A100827 in OEIS).
There are many odd highly cototient numbers. In fact, after 8, all the numbers listed above are odd, and after 167 all the numbers listed above are congruent to 9 modulo 10.
The concept is somewhat analogous to that of highly composite numbers. Just as there are infinitely many highly composite numbers, there are also infinitely many highly cototient numbers. Computations become harder, since integer factorization does, as the numbers get larger.
Read more about Highly Cototient Number: Primes
Famous quotes containing the words highly and/or number:
“There are highly gifted spirits who are always infertile simply because, owing to a weakness in temperament, they are too impatient to wait out their pregnancy to term.”
—Friedrich Nietzsche (1844–1900)
“Among a hundred windows shining
dully in the vast side
of greater-than-palace number such-and-such
one burns
these several years, each night
as if the room within were aflame.”
—Denise Levertov (b. 1923)