cp's OEIS Frontend

A049149 Numbers k such that the Euler totient function phi(k) is squarefree.

%I A049149 #21 Jul 28 2020 22:19:46
%S A049149 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,
%T A049149 98,103,107,118,121,131,134,139,142,158,166,167,179,191,206,211,214,
%U A049149 223,227,239,242,262,263,278,283,311,331,334,347,358,359,367,382,383
%N A049149 Numbers k such that the Euler totient function phi(k) is squarefree.
%C A049149 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
%H A049149 Ivan Neretin, <a href="/A049149/b049149.txt">Table of n, a(n) for n = 1..10000</a>
%H A049149 William D. Banks and Francesco Pappalardi, <a href="https://doi.org/10.1016/j.jnt.2006.01.011">Values of the Euler function free of kth powers</a>, Journal of Number Theory, Vol. 120, No. 2 (2006), pp. 326-348.
%H A049149 Francesco Pappalardi, Filip Saidak and Igor E. Shparlinski, <a href="https://doi.org/10.1016/S0022-314X(03)00110-0">Square-free values of the Carmichael function</a>, Journal of Number Theory, Vol. 103, No. 1 (2003), pp. 122-131.
%F A049149 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
%e A049149 a(17) = 49 is here because phi(49) = 42 = 2*3*7 is squarefree. Primes p, such that p-1 is squarefree are included.
%t A049149 Select[Range[100], MoebiusMu[EulerPhi[#]] != 0 &]
%o A049149 (PARI) isok(n) = issquarefree(eulerphi(n)); \\ _Michel Marcus_, Aug 24 2016
%Y A049149 Cf. A000010, A000720, A005117, A005596, A039787, A013929.
%K A049149 nonn
%O A049149 1,2
%A A049149 _Labos Elemer_
%E A049149 Corrected by _T. D. Noe_, Oct 25 2006