# A discrete mathematics problem by D G

Let $$a_\beta(n)$$ and $$b_\beta(n)$$ equal the number of digits and the number of unique digits in the base $$\beta$$ representation of $$n$$, respectively.

Let $$f_\beta(n) = a_\beta(n) + b_\beta(n)$$.

Let $$r_\beta(g)$$ equal the smallest positive integer $$n$$ such that $$f_\beta(n) = g$$.

Given $$\beta$$, it can be shown that for all integers $$g$$ above some value, $$r_\beta(g) = \beta^{g - \kappa_{\beta}} + \lambda_{\beta}$$, where $$\kappa_{\beta}$$ and $$\lambda_{\beta}$$ are constants.

Find the sum of the digits of $$(\lambda_{100})_{100}$$ ($$\lambda_{100}$$ represented in base $$100$$).

Examples

$$a_{10}(12321) = 5, b_{10}(12321) = 3$$

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