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Sparsely totient number

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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 φ is Euler's totient function. The first few sparsely totient numbers are:


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:

  1. m has a prime factor p ≥ 11, so φ(m) ≥ φ(11) = 10 > φ(18).
  2. m is a multiple of 7 and m/7 ≥ 3, so φ(m) ≥ 2φ(7) = 12 > φ(18).
  3. m is a multiple of 5 and m/5 ≥ 4, so φ(m) ≥ 2φ(5) = 8 > φ(18).
  4. m is a multiple of 3 and m/3 ≥ 7, so φ(m) ≥ 4φ(3) = 8 > φ(18).
  5. 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 Shiu in 1986.


  • 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.


Template:Classes of natural numbers

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