%I
%S 1,1,2,2,4,5,10,15,26,42,74,121,212,357,620,1064,1856,3209,5618,9794,
%T 17192,30153,53114,93554,165308,292250,517802,918207,1630932,2899434,
%U 5161442,9196168,16402764,29281168,52319364,93555601,167427844
%N Exponents f(n), n = 1, 2, ..., for the infinite product 1 z  z^2  z^3 =Product_{n=1}^{\infty} (1z^n)^f(n).
%C Let w = z + z^2 + z^3. Then 1  z  z^2  z^3 = 1  1w = (by the cyclotomic identity) Product_{n=1}^{\infty} (1w^n)^P(1,n), where P is the necklace polynomial. P is a counting function. Is f also a counting function?
%D T. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, 1976, Theorem 14.8.
%F Let r(n) be the coefficient of z^n in 1  z  z^2  z^3, so that r(0) = 1 and r(n) = 0 for n>3. Let F(k) satisfy the recurrence n r(n) + sum_{k=1}^n r(nk)F(k) = 0. Let \mu be the usual M\"obius function. Then f(n) = (1/n) sum_{dn} \mu(n/d) F(d) (so that n*f(n) is the M\"obius inverse of F(n).)
%e f(1) = f(2) = 1 because 1  z  z^2  z^3 = (1z)^1 *(1z^2)^1 * ....
%K nonn
%O 1,3
%A Barry Brent (barrybrent(AT)member.ams.org), Feb 04 2007
