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

1,2

Comments

Consists of 1, 2, 4, p, p^2, 2p, and 2p^2, where p are the odd primes from A039787. - Ivan Neretin, Aug 24 2016

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.

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

Example

a(17) = 49 is here because phi(49) = 42 = 2*3*7 is squarefree. Primes p, such that p-1 is squarefree are included.

Mathematica

Select[Range[100], MoebiusMu[EulerPhi[#]] != 0 &]

Prog

(PARI) isok(n) = issquarefree(eulerphi(n)); \\ Michel Marcus, Aug 24 2016

Crossrefs

Cf. A000010, A000720, A005117, A005596, A039787, A013929.

Sequence in context: A225529 A065156 A097987 * A332555 A304206 A243498

Adjacent sequences: A049146 A049147 A049148 * A049150 A049151 A049152

Keyword

nonn

Author

Labos Elemer

Extensions

Corrected by T. D. Noe, Oct 25 2006

Status

approved