A Unique Function

Algebra Level 4

For non-negative integers a1,a2,a3,,an9a_1, a_2, a_3, \ldots , a_n \leq 9 , let

f(a1,a2,,an)=a1+11a2+111a3++111111Number of 1s=nanf(a_1, a_2, \ldots , a_n) = a_1 + 11a_2 + 111a_3 + \ldots + \underbrace{111 \ldots 111}_{\text{Number of 1s}=n}a_n

Find number of positive integers N<107N \lt 10^7 such that there exists no possible set of non-negative integers a1,a2,a3,,an9a_1, a_2, a_3, \ldots , a_n \leq 9 such that N=f(a1,a2,,an)N = f(a_1, a_2, \ldots , a_n).


All of my problems are original.

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