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A219009
Coefficients of the Dirichlet series for zeta(4s)/zeta(s).
7
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
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
KEYWORD
sign,mult
AUTHOR
Benoit Cloitre, Nov 09 2012
STATUS
approved