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,

where 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 .
  • holds for an exponent .
  • It is conjectured that .

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.
    This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.