A219009 - OEIS

%I #44 May 03 2022 11:05:35

%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,

%T -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,

%U -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

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

%C Different from A197774.

%C Möbius transform of the characteristic function of A000583. - _Amiram Eldar_, May 03 2022

%H Antti Karttunen, <a href="/A219009/b219009.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

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

%F 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

%p Z := proc(n,k)

%p     local a,pf,e ;

%p     a := 1 ;

%p     for pf in ifactors(n)[2] do

%p         e := pf[2] ;

%p         if modp(e,k) = 0 then

%p             ;

%p         elif modp(e,k) = 1 then

%p             a := -a ;

%p         else

%p             a := 0 ;

%p         end if;

%p     end do;

%p     a;

%p end proc:

%p A219009 := proc(n)

%p     Z(n,4) ;

%p end proc: # _R. J. Mathar_, May 28 2016

%t 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 *)

%t 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 *)

%o (PARI) a(n)=sumdiv(n,d,if(issquare(d),issquare(sqrtint(d)),0)*moebius(n/d))

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1-X)/(1-X^4))[n], ", ")) \\ _Vaclav Kotesovec_, Jun 14 2020

%o (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.

%Y 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.

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

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

%K sign,mult

%O 1

%A _Benoit Cloitre_, Nov 09 2012