A049200 - OEIS

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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

(list; graph; refs; listen; history; text; internal format)

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