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Problem 924

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Larger Digit Permutation II

Let $B(n)$ be the smallest number larger than $n$ that can be formed by rearranging digits of $n$, or $0$ if no such number exists. For example, $B(245) = 254$ and $B(542) = 0$.

Define $a_0 = 0$ and $a_n = a_{n - 1}^2 + 2$ for $n > 0$.
Let $\displaystyle U(N) = \sum_{n = 1}^N B(a_n)$. You are given $U(10) \equiv 543870437 \pmod{10^9+7}$.

Find $U(10^{16})$. Give your answer modulo $10^9 + 7$.

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更大的数字重排（二）

令$B(n)$为通过重排$n$的数字所能形成的比$n$大的最小数，如果不存在这样的数则为$0$。例如，$B(245) = 254$，$B(542) = 0$。

定义$a_0 = 0$，且对于$n > 0$有$a_n = a_{n - 1}^2 + 2$。
令$\displaystyle U(N) = \sum_{n = 1}^N B(a_n)$。已知$U(10) \equiv 543870437 \pmod{10^9+7}$。

求$U(10^{16})$，并对$10^9 + 7$取余作为你的答案。

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