A049200 Euler totient function phi applied to the n-th squarefree number.

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, 40, 12, 42, 22, 46, 32, 52, 40, 36, 28, 58, 60, 30, 48, 20, 66, 44, 24, 70, 72, 36, 60, 24, 78, 40, 82, 64, 42, 56, 88, 72, 60, 46, 72, 96, 100, 32, 102, 48, 52, 106, 108, 40, 72

Offset: 1

Author

Labos Elemer

Keywords

• nonn
• easy

Examples

			The 12th squarefree number is 17 and phi(17) is 16, so a(12)=16.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• D. R. Ward, Some Series Involving Euler's Function, Journal of the London Mathematical Society, Vol. 1, No. 4 (1927), pp. 210-214.

Crossrefs

Cf. A000010, A005117, A013929, A083343.

Cf. A001221, A082695, A265668.

Cf. A013661, A065464.

Programs

Haskell

a049200 1 = 1
a049200 n = product $ map (subtract 1) $ a265668_row n
-- Reinhard Zumkeller, Dec 13 2015

Magma

[EulerPhi(n): n in [1..300] | IsSquarefree(n)]; // Vincenzo Librandi, Jul 13 2015

Maple

map(numtheory:-phi,select(numtheory:-issqrfree, [$1..1000])); # Robert Israel, Jul 12 2015

Mathematica

EulerPhi/@Select[Range[200],SquareFreeQ] (* Harvey P. Dale, Jan 13 2015 *)

PARI

lista(nn) = {for(n=1, nn, if (issquarefree(n), print1(eulerphi(n), ", ")));} \\ Michel Marcus, Jul 12 2015

Formula

a(n) = A000010(A005117(n)).

{phi(x) ; abs(mu(x)) = 1}.

a(n) = Product_{k = 1..A001221(n)} (A265668(n,k) + 1). - Reinhard Zumkeller, Dec 13 2015

Sum_{n>=1} 1/(A005117(n)*a(n)) = A082695. - Amiram Eldar, Oct 14 2020

Lim_{n->oo} Sum_{k=1..n} 1/a(k) - log(a(n)) = A083343 (Ward, 1927). - Amiram Eldar, Mar 05 2021

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