If $a_n$ satisfies the necklace congruence, you can verify that $\exp \left( \sum_{k \ge 1} a_k \frac{t^k}{k} \right) = \prod_{n \ge 1} (1-x^n)^{-\frac{b_n}{n}}$, where $b_n$ is an integer sequence given by $b := a * \mu$, and so $b(n) \equiv 0 \mod n$ and the product must have integer coefficients.

Other direction: Assume $\exp \left( \sum_{k \ge 1} a_k \frac{t^k}{k} \right)$ has integral coefficients. Again, one can write it as $\prod_{n \ge 1} (1-x^n)^{-\frac{b_n}{n}}$ where $b = a * \mu$. Because the coefficient of $x^n$ is $\frac{b_n}{n}+$ a sum consisting of binomial coefficients depending on $\{ \frac{b_i}{i} \}_{i<n}$, an induction arguments shows that $\frac{b_n}{n}$ must be integral for all $n$, and so $a_n$ satisfies the necklace congruence.

]]>To fix the issue in your current definition of the map, just sum over i instead of over i,j.

]]>By the way, do you have a reference on this Picard group material that you could point me towards?

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