Consider a number system which does not have the number 7 but has all other numbers. So the numbers are $1,2,3,4,5,6,8,9,10,11,12,13,14,15,16,18,...$. I want to find what is the $10^k$ number, where $k$ is an integer. For example $10^{th}$ number is $11$.
My try: I found that $10^k$ number is $11^k$ if $k \le5$. But for $k=6,11^6=1771561$ which contains $7$. So I am unable to solve this problem. If anyone can help it would be great. Thanks.
Write your number as $a_0+a_1\cdot9^1+a_2\cdot9^2+a_3\cdot9^3+a_4\cdot9^4+a_5\cdot9^5+ \dots $ with $0\le a_i\lt 9$
Then if $0\le a_i\le 6$ set $b_i=a_i$ and for $7\le a_i\le 8$ set $b_i=a_i+1$. Then the expression you are looking for is $$b_rb_{r-1}\dots b_0$$