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A256232 Multiplicative with a(2^e) = 1-e, a(3^e) = 1, a(p^e) = e+1 if p>3.

%I A256232 #17 Sep 29 2024 21:10:58
%S A256232 1,0,1,-1,2,0,2,-2,1,0,2,-1,2,0,2,-3,2,0,2,-2,2,0,2,-2,3,0,1,-2,2,0,2,
%T A256232 -4,2,0,4,-1,2,0,2,-4,2,0,2,-2,2,0,2,-3,3,0,2,-2,2,0,4,-4,2,0,2,-2,2,
%U A256232 0,2,-5,4,0,2,-2,2,0,2,-2,2,0,3,-2,4,0,2,-6
%N A256232 Multiplicative with a(2^e) = 1-e, a(3^e) = 1, a(p^e) = e+1 if p>3.
%H A256232 Reinhard Zumkeller, <a href="/A256232/b256232.txt">Table of n, a(n) for n = 1..10000</a>
%F A256232 Moebius transform is period 6 sequence [1, -1, 0, -1, 1, 0, ...].
%F A256232 G.f.: Sum_{k>0} x^k / (1 + x^k) * Kronecker(9, k).
%F A256232 G.f.: Sum_{k>0} x^k / (1 - x^k) * (-1)^floor(k/3) * Kronecker(-3, k).
%F A256232 G.f.: Sum_{k>0} x^k / (1 - (-x)^k) * (-1)^(k mod 4 = 0) * (k mod 3 > 0).
%F A256232 G.f.: Sum_{k>0} (x^k - x^(2*k)) / (1 + x^(3*k)).
%F A256232 G.f.: Sum_{k>0} (x^k + x^(2*k)) / (1 - x^(3*k)) * (-1)^(k-1).
%F A256232 a(n) = (-1)^(n mod 4 = 0) * A099751(n).
%F A256232 From _Amiram Eldar_, Nov 30 2022: (Start)
%F A256232 Dirichlet g.f.: zeta(s)^2*(1 - 2^(1-s))*(1 - 1/3^s).
%F A256232 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(2)/3 = 0.462098... . (End)
%F A256232 a(n) = Sum_{d|n} A092220(d). - _Ridouane Oudra_, Sep 29 2024
%e A256232 G.f. = x + x^3 - x^4 + 2*x^5 + 2*x^7 - 2*x^8 + x^9 + 2*x^11 - x^12 + ...
%t A256232 a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -3, d] (-1)^Quotient[ d, 3], {d, Divisors@n}]];
%t A256232 a[ n_] := SeriesCoefficient[ Sum[ (x^k - x^(2*k)) / (1 + x^(3*k)), {k, n}], {x, 0, n}];
%o A256232 (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (-1)^(d\3) * kronecker( -3, d)))};
%o A256232 (Haskell)
%o A256232 a099751 n = product $ zipWith f (a027748_row n) (a124010_row n)
%o A256232    where f 2 e = e - 1; f 3 e = 1; f _ e = e + 1
%o A256232 -- _Reinhard Zumkeller_, Mar 20 2015
%Y A256232 Cf. A099751.
%Y A256232 Cf. A027748, A124010, A092220.
%K A256232 sign,mult
%O A256232 1,5
%A A256232 _Michael Somos_, Mar 19 2015