{"app":{"locale":"en-GB"},"problem":{"content":"

Consider the following process that can be applied recursively to any positive integer $n$:

if $n = 1$ do nothing and the process stops,
if $n$ is divisible by 7 divide it by 7,
otherwise add 1.

Define $g(n)$ to be the number of 1's that must be added before the process ends. For example:

$125\\xrightarrow{\\scriptsize{+1}} 126\\xrightarrow{\\scriptsize{\\div 7}} 18\\xrightarrow{\\scriptsize{+1}} 19\\xrightarrow{\\scriptsize{+1}} 20\\xrightarrow{\\scriptsize{+1}} 21\\xrightarrow{\\scriptsize{\\div 7}} 3\\xrightarrow{\\scriptsize{+1}} 4\\xrightarrow{\\scriptsize{+1}} 5\\xrightarrow{\\scriptsize{+1}} 6\\xrightarrow{\\scriptsize{+1}} 7\\xrightarrow{\\scriptsize{\\div 7}} 1$.

Eight 1's are added so $g(125) = 8$. Similarly $g(1000) = 9$ and $g(10000) = 21$.

Define $S(N) = \\sum_{n=1}^{N} g(n)$ and $H(K) = S\\left(\\frac{7^K-1}{11}\\right)$. You are given $H(10) = 690409338$.

Find $H(10^9)$ modulo $1\\,117\\,117\\,717$.

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