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A049200
Euler totient function phi applied to the n-th squarefree number.
5
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
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved