OFFSET
1,2
COMMENTS
The non-principal Dirichlet character mod 3 is chi(n) = A049347(n-1). The twisted Euler phi-function is defined as a(n) = phi(n,chi) = n*Product_{p|n} (1-chi(p)/p), where the product is over all primes p that divide n.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Olivier Bordellès and Benoit Cloitre, An alternating sum involving the reciprocal of certain multiplicative functions, J. Int. Seq. 16 (2013), Article 13.6.3.
Jerzy Kaczorowski and Kazimierz Wiertelak, On the sum of the twisted Euler function, Int. J. Numb. Theory 8 (7) (2012), 1741-1761.
FORMULA
Multiplicative with a(3^e) = 3^e, a(p^e) = p^(e-1)*(p-1) if p == 1 (mod 3) and a(p^e) = p^(e-1)*(p+1) if p == 2 (mod 3). - R. J. Mathar, Jul 10 2013
From Amiram Eldar, Oct 13 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1/(2 * A086724) = 0.639957... . (End)
MAPLE
chi := proc(n)
op(1+(n mod 3), [0, 1, -1]) ;
end proc:
A227128 := proc(n)
local a, p ;
a := n ;
for p in numtheory[factorset](n) do
a := a*(1-chi(p)/p) ;
end do:
a ;
end proc:
MATHEMATICA
f[p_, e_] := If[Mod[p, 3] == 2, p + 1, p - 1]*p^(e - 1); f[3, e_] := 3^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 13 2022 *)
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 3, 3^f[i, 2], f[i, 1]^(f[i, 2] - 1) * (f[i, 1] + (-1)^(f[i, 1]%3))))}; \\ Amiram Eldar, Oct 13 2022
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
R. J. Mathar, Jul 02 2013
STATUS
approved