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A219009
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Coefficients of the Dirichlet series for zeta(4s)/zeta(s).
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7
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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
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OFFSET
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1
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COMMENTS
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LINKS
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FORMULA
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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
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MAPLE
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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:
Z(n, 4) ;
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MATHEMATICA
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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 *)
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PROG
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(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.
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CROSSREFS
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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.
Differs from A197774 for the first time at n=32, where a(32) = -1, while A197774(32) = 0.
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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