Sparsely totient number

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,

\varphi(m)>\varphi(n)

where $\varphi$ is Euler's totient function. The first few sparsely totient numbers are:

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.

Properties

If P(n) is the largest prime factor of n, then $\liminf P(n)/\log n=1$ .
$P(n)\ll \log^\delta n$ holds for an exponent $\delta=37/20$ .
It is conjectured that $\limsup P(n) / \log n = 2$ .

References

Baker, Roger C.; Harman, Glyn (1996). "Sparsely totient numbers". Ann. Fac. Sci. Toulouse, VI. Sér., Math. 5 (2): 183–190. ISSN 0240-2963. Zbl 0871.11060.
Masser, D.W.; Shiu, P. (1986). "On sparsely totient numbers". Pac. J. Math. 121: 407–426. ISSN 0030-8730. MR 0819198. Zbl 0538.10006.

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