login
A227128
The twisted Euler phi-function for the non-principal Dirichlet character mod 3.
2
1, 3, 3, 6, 6, 9, 6, 12, 9, 18, 12, 18, 12, 18, 18, 24, 18, 27, 18, 36, 18, 36, 24, 36, 30, 36, 27, 36, 30, 54, 30, 48, 36, 54, 36, 54, 36, 54, 36, 72, 42, 54, 42, 72, 54, 72, 48, 72, 42, 90, 54, 72, 54, 81, 72, 72, 54, 90, 60, 108, 60, 90, 54, 96
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.
The sequence appears to be the Dirichlet convolution of the sequence A055615(n) and a sequence of signed 1's with the same characteristic function as A156277.
Sequences phi(n,chi) are defined as well for chi=A101455, chi=A080891, chi=A134667 and so on.
LINKS
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)
a(n) = A227128(n)/2 if n divisible by 3, and a(n) = A227128(n) otherwise.
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
Cf. A049347, A080891, A086724, A101455, A134667, A195459 (for the principal character mod 3), A227128.
Sequence in context: A219852 A023842 A165885 * A061795 A110261 A168237
KEYWORD
nonn,easy,mult
AUTHOR
R. J. Mathar, Jul 02 2013
STATUS
approved