

A049149


Numbers k such that the Euler totient function phi(k) is squarefree.


6



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
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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. 326348.
Francesco Pappalardi, Filip Saidak and Igor E. Shparlinski, Squarefree values of the Carmichael function, Journal of Number Theory, Vol. 103, No. 1 (2003), pp. 122131.


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 p1 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



