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A035218 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 36.

%I A035218 #36 Aug 27 2023 04:22:42
%S A035218 1,1,1,1,2,1,2,1,1,2,2,1,2,2,2,1,2,1,2,2,2,2,2,1,3,2,1,2,2,2,2,1,2,2,
%T A035218 4,1,2,2,2,2,2,2,2,2,2,2,2,1,3,3,2,2,2,1,4,2,2,2,2,2,2,2,2,1,4,2,2,2,
%U A035218 2,4,2,1,2,2,3,2,4,2,2,2,1,2,2,2,4,2,2,2,2,2,4,2,2,2,4,1,2,3,2,3,2,2,2,2,4
%N A035218 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 36.
%C A035218 a(n) is the number of factors (over Q) of the polynomial x^(2n) - x^n + 1. - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 30 2003
%C A035218 This sequence is multiplicative. Just as (A001227)(n) is the number of ways to write n as differences of 3-gonal numbers, this sequence is the number of ways to write n as difference of (-1)-gonal numbers. If p_e(n):=1/2*n*((e-2)*n+(4-e)) is the n-th e-gonal number, then 2*a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=-1. - Volker Schmitt (clamsi(AT)gmx.net), Oct 11 2004
%C A035218 a(n) is the number of divisors of n not divisible by 2 or 3. For example, a(36)=1 because 1 is the only such divisor of 36. a(10) = 2 because we count the divisors 1 and 5. - _Geoffrey Critzer_, Feb 15 2015
%H A035218 Antti Karttunen, <a href="/A035218/b035218.txt">Table of n, a(n) for n = 1..65537</a>
%F A035218 a(n) = d(6n) - d(3n) - d(2n) + d(n) where d() is the divisor function. - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 30 2003
%F A035218 Multiplicative with a(2^e)=1, a(3^e)=1, a(p^e)=e+1 if p>3. Inverse Möbius transform is periodic with 1, 0, 0, 0, 1, 0. - Volker Schmitt (clamsi(AT)gmx.net), Oct 11 2004
%F A035218 Dirichlet g.f.: zeta(s)^2*(1 - 1/2^s)*(1 - 1/3^s). - _Geoffrey Critzer_, Feb 15 2015
%F A035218 From _Antti Karttunen_, Oct 03 2018: (Start)
%F A035218 a(n) = A279060(n) + A319995(n).
%F A035218 a(n) = A320015(n) + ch15(n), where ch15 is the characteristic function of numbers of the form +-1 mod 6, i.e., ch15(n) = A232991(n-1).
%F A035218 (End)
%F A035218 Sum_{k=1..n} a(k) ~ n*(log(n) + 2*gamma + log(12)/2 - 1)/3, where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jan 29 2019
%p A035218 res:=1; ifac:=op(ifactors(i))[2]; for pfac in ifac do; if pfac[1]>3 then res:=res*(pfac[2]+1); a(n):=res;
%t A035218 nn = 81; f[list_, i] := list[[i]]; a = Prepend[Drop[Table[Boole[Min[FactorInteger[n][[All, 1]]] > 3], {n, 1, nn}], 1], 1]; b = Table[1, {nn}]; Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* _Geoffrey Critzer_, Feb 15 2015 *)
%t A035218 f[p_, e_] := If[p >= 5, e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Aug 27 2023 *)
%o A035218 (PARI) m=36; direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%o A035218 (PARI) a(n) = sumdiv(n, d, (d % 2) && (d % 3)); \\ _Michel Marcus_, Feb 16 2015
%Y A035218 Cf. A035191, A000005, A001227, A279060, A319995, A320015, A001620.
%K A035218 nonn,mult,easy
%O A035218 1,5
%A A035218 _N. J. A. Sloane_
%E A035218 More terms from _Antti Karttunen_, Oct 03 2018