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%I #40 Oct 09 2023 02:19:42
%S 1,1,2,4,2,6,4,10,12,6,8,16,18,12,10,22,12,28,8,30,20,16,24,36,18,24,
%T 40,12,42,22,46,32,52,40,36,28,58,60,30,48,20,66,44,24,70,72,36,60,24,
%U 78,40,82,64,42,56,88,72,60,46,72,96,100,32,102,48,52,106,108,40,72
%N Euler totient function phi applied to the n-th squarefree number.
%H Robert Israel, <a href="/A049200/b049200.txt">Table of n, a(n) for n = 1..10000</a>
%H D. R. Ward, <a href="https://doi.org/10.1112/jlms/s1-2.4.210">Some Series Involving Euler's Function</a>, Journal of the London Mathematical Society, Vol. 1, No. 4 (1927), pp. 210-214.
%F a(n) = A000010(A005117(n)).
%F {phi(x) ; abs(mu(x)) = 1}.
%F a(n) = Product_{k = 1..A001221(n)} (A265668(n,k) + 1). - _Reinhard Zumkeller_, Dec 13 2015
%F Sum_{n>=1} 1/(A005117(n)*a(n)) = A082695. - _Amiram Eldar_, Oct 14 2020
%F Lim_{n->oo} Sum_{k=1..n} 1/a(k) - log(a(n)) = A083343 (Ward, 1927). - _Amiram Eldar_, Mar 05 2021
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)^2/2) * Product_{p prime} (1 - 2/p^2 + 1/p^3) = A013661^2 * A065464 / 2 = 0.57938048727453660946... . - _Amiram Eldar_, Oct 09 2023
%e The 12th squarefree number is 17 and phi(17) is 16, so a(12)=16.
%p map(numtheory:-phi,select(numtheory:-issqrfree, [$1..1000])); # _Robert Israel_, Jul 12 2015
%t EulerPhi/@Select[Range[200],SquareFreeQ] (* _Harvey P. Dale_, Jan 13 2015 *)
%o (PARI) lista(nn) = {for(n=1, nn, if (issquarefree(n), print1(eulerphi(n), ", ")));} \\ _Michel Marcus_, Jul 12 2015
%o (Magma) [EulerPhi(n): n in [1..300] | IsSquarefree(n)]; // _Vincenzo Librandi_, Jul 13 2015
%o (Haskell)
%o a049200 1 = 1
%o a049200 n = product $ map (subtract 1) $ a265668_row n
%o -- _Reinhard Zumkeller_, Dec 13 2015
%Y Cf. A000010, A005117, A013929, A083343.
%Y Cf. A001221, A082695, A265668.
%Y Cf. A013661, A065464.
%K nonn,easy
%O 1,3
%A _Labos Elemer_
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