Given a natural number, What is the maximal natural number below it, whose sums of digits in base 10 and base 2 are the same? Is there a clever algorithm to do this aside from the brute force search? http://oeis.org/A037308 is the sequence of those numbers.
Related questions are: Does there always exist a natural number the sum of the digits in the binary and decimal representations of which both equal to a given natural number? Is the maximum of such numbers for the given digit sum finite?
We can see that, in the sequence of natural numbers the binary and decimal representations of which are equal, an even number $n$ appears if and only if the odd number $n+1$ does.
I have posted this question in math.stackexchange.com and have not gotten any answer.
What if the upper bound of the number is given as 1000, instead of a general method, is there a trick to solve it quickly?