A227128 The twisted Euler phi-function for the non-principal Dirichlet character mod 3.

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

Author

R. J. Mathar, Jul 02 2013

Keywords

• nonn
• easy
• mult

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

• 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.

Crossrefs

Cf. A049347, A080891, A086724, A101455, A134667, A195459 (for the principal character mod 3), A227128.

Programs

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 *)

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

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)