OFFSET
1,8
COMMENTS
Dirichlet g.f. of b(n) = a(n)/A256689(n) is (zeta (x))^(-1/3).
Denominator is the same as for Dirichlet g.f. (zeta(x))^(+1/3).
Formula holds for general Dirichlet g.f. zeta(x)^(-1/k) with k = 1, 2, ...
LINKS
Wolfgang Hintze, Table of n, a(n) for n = 1..500
FORMULA
with k = 3;
zeta(x)^(-1/k) = Sum_{n>=1} b(n)/n^x;
c(1,n)=b(n); c(k,n) = Sum_{d|n} c(1,d)*c(k-1,n/d), k>1;
Then solve c(k,n) = mu(n) for b(m);
a(n) = numerator(b(n)).
MATHEMATICA
k = 3;
c[1, n_] = b[n];
c[k_, n_] := DivisorSum[n, c[1, #1]*c[k - 1, n/#1] & ]
nn = 100; eqs = Table[c[k, n]==MoebiusMu[n], {n, 1, nn}];
sol = Solve[Join[{b[1] == 1}, eqs], Table[b[i], {i, 1, nn}], Reals];
t = Table[b[n], {n, 1, nn}] /. sol[[1]];
num = Numerator[t] (* A257099 *)
den = Denominator[t] (* A256689 *)
CROSSREFS
KEYWORD
sign
AUTHOR
Wolfgang Hintze, Apr 16 2015
STATUS
approved