A219009 Coefficients of the Dirichlet series for zeta(4s)/zeta(s).

1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, 1, -1, 0, -1, 0, 1, 1, -1, 0, 0, 1, 0, 0, -1, -1, -1, -1, 1, 1, 1, 0, -1, 1, 1, 0, -1, -1, -1, 0, 0, 1, -1, -1, 0, 0, 1, 0, -1, 0, 1, 0, 1, 1, -1, 0, -1, 1, 0, 0, 1, -1, -1, 0, 1, -1, -1, 0, -1, 1, 0, 0, 1, -1, -1, -1, 1, 1, -1, 0, 1, 1, 1, 0, -1, 0, 1, 0, 1, 1, 1, 1, -1

Offset

1

Comments

Different from A197774.

Möbius transform of the characteristic function of A000583. - Amiram Eldar, May 03 2022

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n

Formula

G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = Sum_{n>=1} x^(n^4). - Geoffrey Critzer, Mar 20 2015

Multiplicative with a(p^e) = (-1)^e if e == {0, 1} mod 4, and 0 if e == {2, 3} mod 4. [deduced from R. J. Mathar's Maple-program] - Antti Karttunen, May 03 2022

Maple

Z := proc(n, k)
    local a, pf, e ;
    a := 1 ;
    for pf in ifactors(n)[2] do
        e := pf[2] ;
        if modp(e, k) = 0 then
            ;
        elif modp(e, k) = 1 then
            a := -a ;
        else
            a := 0 ;
        end if;
    end do;
    a;
end proc:
A219009 := proc(n)
    Z(n, 4) ;
end proc: # R. J. Mathar, May 28 2016

Mathematica

nn = 100; f[x_] := Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Sum[x^(n^4), {n, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten (* Geoffrey Critzer, Mar 20 2015 *)
f[p_, e_] := If[Mod[e, 4] < 2, (-1)^e, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 03 2022, after Maple-program *)

Prog

(PARI) a(n)=sumdiv(n, d, if(issquare(d), issquare(sqrtint(d)), 0)*moebius(n/d))
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1-X)/(1-X^4))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020
(PARI) A219009(n) = { my(f=factor(n)); prod(k=1, #f~, if((f[k, 2]%4)>1, 0, ((-1)^f[k, 2]))); }; \\ Antti Karttunen, May 03 2022, after Maple-program.

Crossrefs

Absolute values of these terms is given by A353519, which is the characteristic function of A252895. The complement of the latter, A252849, gives the positions of zeros.

Cf. A000583, A008683, A008836, A046951, A063775, A210826, A253206, A307430 (Dirichlet inverse).

Differs from A197774 for the first time at n=32, where a(32) = -1, while A197774(32) = 0.

Sequence in context: A323154 A060038 A197774 * A353519 A267708 A167021

Adjacent sequences: A219006 A219007 A219008 * A219010 A219011 A219012

Keyword

sign,mult

Author

Benoit Cloitre, Nov 09 2012

Status

approved