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A227128 The twisted Euler phi-function for the non-principal Dirichlet character mod 3. 1
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..64.

J. Kaczorowski, K. Wiertelak, On the sum of the twisted Euler function, Int. J. Numb. Theory 8 (7) (2012) 1741-1761

O. Bordelles, B. Cloitre, An alternating sum involving the reciprocal of certain multiplicative functions, J. Int. Seq. 16 (2013) Article 13.6.3

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

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:

CROSSREFS

Cf. A195459 (for the principal character mod 3).

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

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Last modified July 22 03:22 EDT 2017. Contains 289648 sequences.