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A035191 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 9.

%I A035191 #51 Oct 03 2024 13:21:53
%S A035191 1,2,1,3,2,2,2,4,1,4,2,3,2,4,2,5,2,2,2,6,2,4,2,4,3,4,1,6,2,4,2,6,2,4,
%T A035191 4,3,2,4,2,8,2,4,2,6,2,4,2,5,3,6,2,6,2,2,4,8,2,4,2,6,2,4,2,7,4,4,2,6,
%U A035191 2,8,2,4,2,4,3,6,4,4,2,10,1
%N A035191 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 9.
%C A035191 Number of divisors of n not congruent to 0 mod 3. - _Vladeta Jovovic_, Oct 26 2001
%C A035191 a(n) is the number of factors (over Q) of the polynomial x^(2n) + x^n + 1 . a(n) = d(3n) - d(n) where d() is the divisor function. - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003
%C A035191 Equals Mobius transform of A011655. - _Gary W. Adamson_, Apr 24 2009
%H A035191 Reinhard Zumkeller, <a href="/A035191/b035191.txt">Table of n, a(n) for n = 1..10000</a>
%F A035191 Multiplicative with a(3^e)=1 and a(p^e)=e+1 for p<>3.
%F A035191 G.f.: Sum_{k>0} x^k*(1+x^k)/(1-x^(3*k)). - _Vladeta Jovovic_, Dec 16 2002
%F A035191 a(n) = A001817(n) + A001822(n). [_Reinhard Zumkeller_, Nov 26 2011]
%F A035191 a(n) = tau(3*n) - tau(n). - _Ridouane Oudra_, Sep 05 2020
%F A035191 From _Amiram Eldar_, Nov 27 2022: (Start)
%F A035191 Dirichlet g.f.: zeta(s)^2 * (1 - 1/3^s).
%F A035191 Sum_{k=1..n} a(k) ~ (2*n*log(n) + (4*gamma + log(3) - 2)*n)/3, where gamma is Euler's constant (A001620). (End)
%F A035191 a(n) = Sum_{d|n} Kronecker(9, d). - _Amiram Eldar_, Nov 20 2023
%F A035191 a(n) = A000005(A038502(n)). - _Ridouane Oudra_, Sep 30 2024
%p A035191 for n from 1 to 500 do a := ifactors(n):s := 1:for k from 1 to nops(a[2]) do p := a[2][k][1]:e := a[2][k][2]: if p=3 then b := 1:else b := e+1:fi:s := s*b:od:printf(`%d,`,s); od:
%p A035191 # alternative
%p A035191 A035191 := proc(n)
%p A035191     A001817(n)+A001822(n) ;
%p A035191 end proc:
%p A035191 [seq(A035191(n),n=1..100)] ; # _R. J. Mathar_, Sep 25 2017
%t A035191 a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[9, #] &]]; Table[ a[n], {n, 1, 100}] (* _G. C. Greubel_, Apr 27 2018 *)
%t A035191 f[3, e_] := 1; f[p_, e_] := e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 26 2020 *)
%o A035191 (PARI) my(m=9); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%o A035191 (PARI) a(n) = sumdiv(n, d, kronecker(9, d)); \\ _Amiram Eldar_, Nov 20 2023
%o A035191 (Haskell)
%o A035191 a035191 n = a001817 n + a001822 n  -- _Reinhard Zumkeller_, Nov 26 2011
%o A035191 (Magma) [NumberOfDivisors(n)/Valuation(3*n, 3): n in [1..100]]; // _Vincenzo Librandi_, Jun 03 2019
%Y A035191 Cf. A011655, A035207, A046913, A054584.
%Y A035191 Cf. A000005, A001227, A001620, A001817, A001817, A038502.
%K A035191 nonn,mult,easy
%O A035191 1,2
%A A035191 _N. J. A. Sloane_