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

1,2

Comments

The non-principal Dirichlet character mod 3 is chi = A102283. 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.

From Jianing Song, May 05 2026: (Start)

For n > 1, we have h(-3*n^2), the class number of the order of Z[(1+sqrt(-3))/2] with discriminant -3*n^2, is equal to (1/3)*a(n).

In general, let d < 0 be a fundamental discriminant and n be a positive integer such that d*n^2 < -4, then h(d*n^2) = L(0,kronecker(d,.)) * n * Product_{p|n, p prime} (1 - kronecker(d,p)/p). This is a combination of Theorem 7.24 in "Primes of the form x^2+ny^2" by Cox and the Dirichlet class number formula. Here L(0,kronecker(-3,.)) = 1/3.

For example, given L(0,kronecker(-3,.)) = 1/3, we have: h(-3*2^2) = (1/3)*a(2) = 1, h(-3*3^2) = (1/3)*a(3) = 1, h(-3*4^2) = (1/3)*a(4) = 2, ... (End)

References

D. A. Cox, Primes of the form x^2+ny^2, Wiley, New York, 1989, p. 146.

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)

a(n) = A227501(n)/2 if n divisible by 3, and a(n) = A227501(n) otherwise.

Sum_{k=1..n} a(k) ~ c * n^2, where c = 1/(2 * A086724) = 0.639957... . (End) [A-number corrected by Jianing Song, May 05 2026]

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. Dirichlet characters: A102283, A101455, A080891, A134667.

Cf. A086724, A195459 (for the principal character mod 3), A227501.

Cf. for fundamental discriminants -3..-24: this sequence, A395768, A395771, A395769, A395772, A395773, A395774, A395775, A395776, A395777.

Sequence in context: A219852 A023842 A165885 * A061795 A110261 A168237

Adjacent sequences: A227125 A227126 A227127 * A227129 A227130 A227131

Keyword

nonn,easy,mult

Author

R. J. Mathar, Jul 02 2013

Status

approved