A Unique Function

Seeing $\displaystyle a_1, a_2 \ldots a_n \leq 9$ reminds of digits. Let $\displaystyle X = \overline{a_na_{n-1}a_{n-2}\ldots a_2a_1}$

Now,

$f(a_1, a_2, \ldots , a_n) = 1\cdot a_1 + 11\cdot a_2 + 111\cdot a_3 + \ldots + \underbrace{111\ldots111}_{\text{Number of 1s = n }}\cdot a_n$

$f(a_1, a_2, \ldots , a_n) = (10^{n-1}a_n + 10^{n-2}a_{n-1} + \ldots + 10^0a_1) + (10^{n-2}a_n + 10^{n-3}a_{n-1} + \ldots + 10^0a_2) +(10^{n-3}a_n + 10^{n-4}a_{n-1} + \ldots + 10^0a_1) + \ldots + (10a_n + a_{n-1}) + a_n$

$f(a_1, a_2, \ldots , a_n) = \overline{a_na_{n-1}\ldots a_1} + \overline{a_na_{n-1}\ldots a_2} + \overline{a_na_{n-1}\ldots a_3} + \ldots + \overline{a_na_{n-1}} + a_n$

Therefore, we to find number of positive integers $\displaystyle X = \overline{a_na_{n-1}\ldots a_1}$ such that $\displaystyle N = \overline{a_na_{n-1}\ldots a_1} + \overline{a_na_{n-1}\ldots a_2} + \overline{a_na_{n-1}\ldots a_3} + \ldots + \overline{a_na_{n-1}} + a_n \lt 10^7$

Now, smallest $X = 0$ and biggest $X = 9 × 10^6$ .

So, $\text{number of integers } = 10^7 - (9× 10^6 + 1)$ N

$\boxed{\text{number of integers } = 999999}$

Notice that $a_1$ is never large enough to reach $11$ , and $a_1+11a_2$ is never large enough to reach $111$ , and so on. This means that we cannot rewrite the sum $a_1+11a_2+\dots+111\dots111a_n$ using different values of $a_1,\dots,a_n$ to sum to the same value. Thus, every combination of values of $a_1,\dots,a_n$ gives a unique value for $f(a_1,\dots,a_n)$ .

So, we simply need the number of combinations of $a_1,\dots,a_n$ such that $0<f(a_1,\dots,a_n)<10^7=10,000,000$ . First note that any nonzero terms past $n=7$ will be larger than $10^7$ , so we are only interested in $a_1,\dots,a_7$ . If $a_7$ is equal to 9, then all the other terms must be zero for the sum to equal 9,999,999 and be less than 10^7. Otherwise, the maximum possible value of the sum will be attained at $a_1=a_2=\dots=a_6=9$ , and $a_7=8$ , for a total of $9+99+999+9999+99999+999999+8888888=9999992<10^7$ Last of all, we require the sum to be greater than zero, so the combination $a_1=\dots=a_7=0$ does not count. Hence, the total number of values of $f(a_1,\dots,a_n)$ less than 10^7 is $\underbrace{10^6\times9}_{\text{10 options for each }a_k\text{ except }a_7}\qquad\underbrace{+1}_{\text{for }9999999}\qquad\underbrace{-1}_{\text{for }0}\qquad=9000000$

There are $10^7-1$ positive integers less than $10^7$ , so the number of positive integers not of this form is $10^7-1-9000000=999999$