Problem 467

Superinteger

An integer s is called a superinteger of another integer n if the digits of n form a subsequence of the digits of s.
For example, 2718281828 is a superinteger of 18828, while 314159 is not a superinteger of 151.

Let p(n) be the nth prime number, and let c(n) be the nth composite number. For example, p(1) = 2, p(10) = 29, c(1) = 4 and c(10) = 18.
{p(i) : i ≥ 1} = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …}
{c(i) : i ≥ 1} = {4, 6, 8, 9, 10, 12, 14, 15, 16, 18, …}

Let P^D the sequence of the digital roots of {p(i)} (C^D is defined similarly for {c(i)}):
P^D = {2, 3, 5, 7, 2, 4, 8, 1, 5, 2, …}
C^D = {4, 6, 8, 9, 1, 3, 5, 6, 7, 9, …}

Let P_n be the integer formed by concatenating the first n elements of P^D (C_n is defined similarly for C^D).
P₁₀ = 2357248152
C₁₀ = 4689135679

Let f(n) be the smallest positive integer that is a common superinteger of P_n and C_n.
For example, f(10) = 2357246891352679, and f(100) mod 1 000 000 007 = 771661825.

Find f(10 000) mod 1 000 000 007.

超级整数

整数s被称为是另一个整数n的超级整数，当且仅当n的数字串是s的数字串的子序列。
例如2718281828是18828的超级整数，而314159不是151的超级整数。

记p(n)是第n个质数，而c(n) 是第n个合数。例如p(1) = 2，p(10) = 29，c(1) = 4，而c(10) = 18。
{p(i) : i ≥ 1} = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …}
{c(i) : i ≥ 1} = {4, 6, 8, 9, 10, 12, 14, 15, 16, 18, …}

记P^D 是{p(i)}中各项的数字根组成的序列（C^D同理）：
P^D = {2, 3, 5, 7, 2, 4, 8, 1, 5, 2, …}
C^D = {4, 6, 8, 9, 1, 3, 5, 6, 7, 9, …}

记P_n是将P^D的前n项连接起来所得的整数（C_n同理）。
P₁₀ = 2357248152
C₁₀ = 4689135679

记f(n)是P_n和C_n的最小公共超级整数。
例如，f(10) = 2357246891352679，而f(100) mod 1 000 000 007 = 771661825。

求f(10 000) mod 1 000 000 007。