Sparsely totient number explained

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, ... .

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 . Glyn Harman . Sparsely totient numbers . Ann. Fac. Sci. Toulouse, VI. Sér., Math. . 5 . 2 . 183–190 . 1996 . 10.5802/afst.826 . 0240-2963 . 0871.11060 . free .
• Masser . D.W. . David Masser . Shiu . P. . On sparsely totient numbers . Pac. J. Math. . 121 . 407–426 . 1986 . 2 . 10.2140/pjm.1986.121.407 . 0030-8730 . 0538.10006 . 819198 . 55350630 . free .