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,

[math]\displaystyle{ \varphi(m)\gt \varphi(n) }[/math]

where [math]\displaystyle{ \varphi }[/math] 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).

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 [math]\displaystyle{ \liminf P(n)/\log n=1 }[/math].
• [math]\displaystyle{ P(n)\ll \log^\delta n }[/math] holds for an exponent [math]\displaystyle{ \delta=37/20 }[/math].
• It is conjectured that [math]\displaystyle{ \limsup P(n) / \log n = 2 }[/math].

References

• Baker, Roger C.; Harman, Glyn (1996). "Sparsely totient numbers". Ann. Fac. Sci. Toulouse, VI. Sér., Math. 5 (2): 183–190. doi:10.5802/afst.826. ISSN 0240-2963. https://eudml.org/doc/73381. 
• Masser, D.W.; Shiu, P. (1986). "On sparsely totient numbers". Pac. J. Math. 121 (2): 407–426. doi:10.2140/pjm.1986.121.407. ISSN 0030-8730. http://projecteuclid.org/euclid.pjm/1102702441.