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%I #36 Feb 03 2021 05:08:22
%S 1,-1,2,0,-1,-2,-1,0,6,1,-1,0,-1,1,-2,0,-1,-6,-1,0,-2,1,-1,0,0,1,18,0,
%T -1,2,-1,0,-2,1,1,0,-1,1,-2,0,-1,2,-1,0,-6,1,-1,0,0,0,-2,0,-1,-18,1,0,
%U -2,1,-1,0,-1,1,-6,0,1,2,-1,0,-2,-1,-1,0,-1,1,0,0,1,2,-1,0,54,1,-1,0,1,1,-2,0,-1,6,1,0,-2,1,1,0,-1,0,-6,0,-1,2,-1,0,2
%N Sum_{d|n} a(d) = n for n = 3^m (m >= 0) and for other n the sum is zero; i.e., the Möbius transform of [1, 0, 3, 0, 0, 0, 0, 0, 9, 0,...].
%C From _Petros Hadjicostas_, Jul 26 2020: (Start)
%C For p prime >= 2, Petrogradsky (2003) defined the multiplicative functions 1_p and mu_p in the following way:
%C 1_p(n) = 1 when gcd(n,p) = 1 and 1_p(n) = 1 - p when gcd(n,p) = p;
%C mu_p(n) = mu(n) when gcd(n,p) = 1 and mu_p(n) = mu(m)*(p^s - p^(s-1)) when n = m*p^s with gcd(m,p) = 1 and s >= 1.
%C We have 1_2(n) = A062157(n), 1_3(n) = A061347(n), A067856(n) = mu_2(n), and a(n) = mu_3(n) for n >= 1.
%C Some of the results by other contributors here and in A067856 can be generalized:
%C (i) Rogel's (1897) formula for A067856 becomes Sum_{d | n} 1_p(d) * mu_p(n/d) = 0 for n > 1. Thus, 1_p is the Dirichlet inverse of mu_p.
%C (ii) _R. J. Mathar_'s Dirichlet g.f. for mu_p becomes 1/(zeta(s) * (1 - p^(1-s))). The Dirichlet g.f. for 1_p is zeta(s) * (1 - p^(1-s)).
%C (iii) _Benoit Cloitre_'s formula becomes 1 = Sum_{k=1..n} mu_p(k)*g_p(n/k), where g_p(x) = floor(x) - p*floor(x/p) = floor(x) mod p.
%C (iv) _Paul D. Hanna_'s formula becomes Sum_{n >= 1} (mu_p(n)/n)*log((1 - x^(n*p))/(1 - x^n)) = x.
%C (v) The definition in the name of the sequence a(n) generalizes to Sum_{d | n} mu_p(d) = n, if n = p^s for s >= 0, and = 0, otherwise. Thus, mu_p(n) = Sum_{p^k | n, k >= 0} mu(n/p^k)*p^k. That is, (mu_p(n): n >= 1) is the Möbius transform of the sequence (b_p(n): n >= 1), where b_p(n) = p^k, if n = p^k for k >= 0, and b_p(n) = 0, otherwise.
%C (vi) We have the Lambert series Sum_{n >= 1} mu_p(n)*x^n/(1 - x^n) = Sum_{k >= 0} p^k*x^(p^k) = x + p*x^p + p^2*x^(p^2) + ..., which generalizes one of the formulas by _Peter Bala_ in A067856.
%C (vii) By differentiating both sides of (iv) w.r.t. x and multiplying both sides by x, we get Sum_{n >= 1} mu_p(n)*(x^n + 2*x^(2*n) + ... + (p-1)*x^(n*(p-1)))/(1 + x^n + x^(2*n) + ... + x^(n*(p-1))) = x, which generalizes another one of _Peter Bala_'s formulas in A067856. It can be thought as a "generalized Lambert series".
%C (viii) Dividing both sides of (vi) by x and integrating w.r.t. x from 0 to y, we get -Sum_{n >= 1} (mu_p(n)/n)*log(1 - y^n) = Sum_{k >= 0} y^(p^k) = y + y^p + y^(p^2) + y^(p^3) + ...
%C (ix) Obviously, f(n) = Sum_{d | n} 1_p(n/d)*g(d) if and only if g(n) = Sum_{d | n} mu_p(n/d)*f(d). (End)
%H Peter Bala, <a href="/A067856/a067856_1.pdf">A signed Dirichlet product of arithmetical functions</a>.
%H V. M. Petrogradsky, <a href="https://doi.org/10.1016/S0196-8858(02)00533-X">Witt's formula for restricted Lie algebras</a>, Advances in Applied Mathematics, 30 (2003), 219-227.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MoebiusTransform.html">Möbius Transform</a>.
%F G.f.: x = Sum_{n >= 1} (a(n)/n)*log(1 + x^n + x^(2*n)).
%F 1 = Sum_{k=1..n} a(k)*g(n/k), where g(x) = floor(x) - 3*floor(x/3). [_Benoit Cloitre_, Nov 11 2010]
%F From _Petros Hadjicostas_, Jul 26 2020: (Start)
%F a(n) = Sum_{3^k | n, k >= 0} mu(n/3^k)*3^k.
%F Dirichlet g.f.: 1/(zeta(s)*(1 - 3^(1-s))).
%F The sequence is the Dirichlet inverse of A061347.
%F Sum_{n >= 1} a(n)*x^n/(1 - x^n) = x + 3*x^3 + 9*x^9 + 27*x^27 + 81*x^81 + ...
%F Sum_{n >= 1} a(n)*(x^n + 2*x^(2*n))/(1 + x^n + x^(2*n)) = x.
%F -Sum_{n >= 1} (a(n)/n)*log(1 - x^n) = x + x^3 + x^9 + x^27 + x^81 + ... (End)
%o (PARI) {a(n)=if(n==1,1,-n*polcoeff(x+sum(k=1,n-1,a(k)/k*subst(log(1+x+x^2+x*O(x^n)),x,x^k+x*O(x^n))),n))}
%Y Cf. A061347, A062157, A067856.
%K sign
%O 1,3
%A _Paul D. Hanna_, Apr 08 2006
%E Offset changed to 1 by _Petros Hadjicostas_, Jul 26 2020
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