Sparsely totient number
In mathematics, a sparsely totient number is a certain kind of even natural number. A natural number, n, is sparsely totient if for all m > n,
- φ(m)>φ(n),
where φ 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 (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) ≥ &phi(32) = 16 > φ(18).
References
- Roger C. Baker & Glyn Harman, "Sparsely totient numbers," Annales de la faculte des sciences de Toulouse Ser. 6 5 no. 2 (1996): 183 - 190
- D. W. Masser & P. Shiu, "On sparsely totient numbers," Pacific J. Math. 121, no. 2 (1986): 407 - 426.