OFFSET
1,7
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
FORMULA
Moebius transform is period 18 sequence [ 1, -2, -2, 2, -1, 4, 1, -2, 0, 2, -1, -4, 1, -2, 2, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = -1 if e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
G.f.: Sum_{k>0} F(x^(6*k - 5)) - F(x^(6*k - 3)) + F(x^(6*k - 1)) where F(x) := x / (1 + x + x^2).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(18*sqrt(3)) = 0.100766631346... . - Amiram Eldar, Nov 23 2023
EXAMPLE
G.f. = x - x^2 - x^3 + x^4 + x^6 + 2*x^7 - x^8 - x^9 - x^12 + 2*x^13 + ...
MATHEMATICA
PROG
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, (-1)^e, p==3, -1, p%6==1, e+1, 1-e%2)))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^9 + A)^3 / eta(x^3 + A) - x * eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^18 + A)^4 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^9 + A)^2), n))};
CROSSREFS
KEYWORD
sign,easy,mult
AUTHOR
Michael Somos, Sep 04 2015
STATUS
approved