Logarithmic Fibonacci ? Sounds Crazy !

Fibonacci sequence is defined as \(F_0=0,F_1=1\) and for \(n\geq 2\), \[F_n=F_{n-1}+F_{n-2}\]

Thus, the fibonacci sequence is \(0,1,1,2,3,5,8,13,...\)

Find the sum of all the Fibonacci numbers \(F_n\) less than \(\textbf{1 billion}\) which follow that \(\log_{10}(F_n) \in \mathbb{Z}\)

Details and assumptions:-

\(\bullet \quad F_n\) denotes \(n^{th}\) number in the Fibonacci sequence.

\(\bullet\quad \log_{10}( F_n) \in \mathbb{Z}\) means that \(F_n\) is of the form \(10^k\) for integer value of \(k\).

This problem is a part of the set Crazy Fibonacci


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