A049149 Numbers k such that the Euler totient function phi(k) is squarefree.
1, 2, 3, 4, 6, 7, 9, 11, 14, 18, 22, 23, 31, 43, 46, 47, 49, 59, 62, 67, 71, 79, 83, 86, 94, 98, 103, 107, 118, 121, 131, 134, 139, 142, 158, 166, 167, 179, 191, 206, 211, 214, 223, 227, 239, 242, 262, 263, 278, 283, 311, 331, 334, 347, 358, 359, 367, 382, 383
Offset: 1
Keywords
Examples
a(17) = 49 is here because phi(49) = 42 = 2*3*7 is squarefree. Primes p, such that p-1 is squarefree are included.
Links
- Ivan Neretin, Table of n, a(n) for n = 1..10000
- William D. Banks and Francesco Pappalardi, Values of the Euler function free of kth powers, Journal of Number Theory, Vol. 120, No. 2 (2006), pp. 326-348.
- Francesco Pappalardi, Filip Saidak and Igor E. Shparlinski, Square-free values of the Carmichael function, Journal of Number Theory, Vol. 103, No. 1 (2003), pp. 122-131.
Programs
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Mathematica
Select[Range[100], MoebiusMu[EulerPhi[#]] != 0 &]
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PARI
isok(n) = issquarefree(eulerphi(n)); \\ Michel Marcus, Aug 24 2016
Formula
The number of terms not exceeding k is (3*a/2) * pi(k) + O(k/(log(k)^c)), where pi(k) = A000720(k), c is any constant > 0, and a = 0.373955... is Artin's constant (A005596) (Pappalardi et al., 2003; Banks and Pappalardi, 2006). - Amiram Eldar, Jul 28 2020
Extensions
Corrected by T. D. Noe, Oct 25 2006
Comments