# sparsely totient number

A sparsely totient number is the largest integer with a given totient. That is, given an sparsely totient number $n$, the inequality $\phi(m)>\phi(n)$ (with $\phi(x)$ being Euler’s totient function) is true for any $m>n$. For example, $\phi(12)=4$. We can then verify that $\phi(13)=12$, $\phi(14)=6$ and $\phi(15)=8$. Accepting as true that the inequality $\phi(n)>\sqrt{n}$ holds for all $n>6$, we don’t need to check any larger numbers to confirm that 12 is the largest integer with 4 as its totient.

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, etc., listed in A036913 of Sloane’s OEIS.

All sparsely totient numbers are even. In 1986, Masser and Shiu proved that the $i$th primorial multiplied by the $i$th prime (for $i>1$, thus: 18, 150, 1470, 25410, 390390, etc.) is a sparsely totient number.

## References

• 1 Roger C. Baker & Glyn Harman, “Sparsely totient numbers,” Annales de la faculte des sciences de Toulouse Ser. 6 5 no. 2 (1996): 183 - 190
• 2 D. W. Masser & P. Shiu, “On sparsely totient numbers,” Pacific J. Math. 121, no. 2 (1986): 407 - 426.
Title sparsely totient number SparselyTotientNumber 2013-03-22 16:48:59 2013-03-22 16:48:59 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Definition msc 11A25