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

1,3

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.

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

Example

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

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 *)

Prog

(PARI) lista(nn) = {for(n=1, nn, if (issquarefree(n), print1(eulerphi(n), ", "))); } \\ Michel Marcus, Jul 12 2015
(Magma) [EulerPhi(n): n in [1..300] | IsSquarefree(n)]; // Vincenzo Librandi, Jul 13 2015
(Haskell)
a049200 1 = 1
a049200 n = product $ map (subtract 1) $ a265668_row n
-- Reinhard Zumkeller, Dec 13 2015

Crossrefs

Cf. A000010, A005117, A013929, A083343.

Cf. A001221, A082695, A265668.

Cf. A013661, A065464.

Sequence in context: A322071 A176342 A340148 * A347100 A164701 A325965

Adjacent sequences: A049197 A049198 A049199 * A049201 A049202 A049203

Keyword

nonn,easy

Author

Labos Elemer

Status

approved