1, 5, 2, 7, 1, has signature \*, \*, 9, 1, 3, 1, 9. We placed stars at the first two places, because we do not know the preceding members of the sequence and, hence, do not know the powers.

Given an $n$-free Fibonacci sequence with a given signature and a given set of remainders, we can build many other sequences with the same signature and the set of remainders.

\[thm:adjustement\] Suppose an $n$-free Fibonacci sequence $s_1$ starts with $a_1$ and $a_2$ that are not divisible by $n$ and the product of the numbers that we divide by while calculating the first $k$ terms is strictly less than $n^m$. Consider an $n$-free Fibonacci sequence $s_2$ that starts with $b_1=a_1+ d_1 n^m$ and $b_2=a_2+ d_2 n^m$ for any integers $d_1$ and $d_2$. The first $k$ terms of both sequences have the same signature and the same set of remainders modulo $n$.

The initial terms of both sequences have the same remainders modulo $n^m$. Hence their sums have the same remainders. The first time we need to divide, we divide by the same power of $n$, say $m_1$, and the result will have the same remainders modulo $n^{m-m_1}$. The next time we 