note on quotients of lacunary functions

Define:

\lambda_{a}(q) \equiv 1 + \sum_{n=1}^{\infty} q^{a^{n}} and

 \l_{a}(q) \equiv 1 + \sum_{n=1}^{\infty} (-1)^{n} q^{a^{n}}

Preliminary numerical experiments suggest that the power series representations of their quotients l_{b}(q)/\lambda_{a}(q) and \lambda_{a}(q)/l_{b}(q) always have integer coefficients. 

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