In mathematics, a sparsely totient number is a certain kind of natural number. A natural number, n, is sparsely totient if for all m > n,
where
2, 6, 12, 18, 30, 42, 60, 66, 90, 120, 126, 150, 210, 240, 270, 330, 420, 462, 510, 630, 660, 690, 840, 870, 1050, 1260, 1320, 1470, 1680, 1890, 2310, 2730, 2940, 3150, 3570, 3990, 4620, 4830, 5460, 5610, 5670, 6090, 6930, 7140, 7350, 8190, 9240, 9660, 9870, ... (sequence A036913 in the OEIS).
For example, 18 is a sparsely totient number because ϕ(18) = 6, and any number m > 18 falls into at least one of the following classes:
- m has a prime factor p ≥ 11, so ϕ(m) ≥ ϕ(11) = 10 > ϕ(18).
- m is a multiple of 7 and m/7 ≥ 3, so ϕ(m) ≥ 2ϕ(7) = 12 > ϕ(18).
- m is a multiple of 5 and m/5 ≥ 4, so ϕ(m) ≥ 2ϕ(5) = 8 > ϕ(18).
- m is a multiple of 3 and m/3 ≥ 7, so ϕ(m) ≥ 4ϕ(3) = 8 > ϕ(18).
- m is a power of 2 and m ≥ 32, so ϕ(m) ≥ ϕ(32) = 16 > ϕ(18).
The concept was introduced by David Masser and Peter Man-Kit Shiu in 1986. As they showed, every primorial is sparsely totient.