OFFSET
1,3
COMMENTS
From Petros Hadjicostas, Jul 26 2020: (Start)
For p prime >= 2, Petrogradsky (2003) defined the multiplicative functions 1_p and mu_p in the following way:
1_p(n) = 1 when gcd(n,p) = 1 and 1_p(n) = 1 - p when gcd(n,p) = p;
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.
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.
Some of the results by other contributors here and in A067856 can be generalized:
(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.
(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)).
(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.
(iv) Paul D. Hanna's formula becomes Sum_{n >= 1} (mu_p(n)/n)*log((1 - x^(n*p))/(1 - x^n)) = x.
(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.
(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.
(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".
(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) + ...
(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)
LINKS
Peter Bala, A signed Dirichlet product of arithmetical functions.
V. M. Petrogradsky, Witt's formula for restricted Lie algebras, Advances in Applied Mathematics, 30 (2003), 219-227.
Eric Weisstein's World of Mathematics, Möbius Transform.
FORMULA
G.f.: x = Sum_{n >= 1} (a(n)/n)*log(1 + x^n + x^(2*n)).
1 = Sum_{k=1..n} a(k)*g(n/k), where g(x) = floor(x) - 3*floor(x/3). [Benoit Cloitre, Nov 11 2010]
From Petros Hadjicostas, Jul 26 2020: (Start)
a(n) = Sum_{3^k | n, k >= 0} mu(n/3^k)*3^k.
Dirichlet g.f.: 1/(zeta(s)*(1 - 3^(1-s))).
The sequence is the Dirichlet inverse of A061347.
Sum_{n >= 1} a(n)*x^n/(1 - x^n) = x + 3*x^3 + 9*x^9 + 27*x^27 + 81*x^81 + ...
Sum_{n >= 1} a(n)*(x^n + 2*x^(2*n))/(1 + x^n + x^(2*n)) = x.
-Sum_{n >= 1} (a(n)/n)*log(1 - x^n) = x + x^3 + x^9 + x^27 + x^81 + ... (End)
PROG
(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))}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Apr 08 2006
EXTENSIONS
Offset changed to 1 by Petros Hadjicostas, Jul 26 2020
STATUS
approved