Phinary representation
Public  04/30/15  10xp  Math  100.0% 
$\phi = \frac{1+\sqrt{5}}{2}$ is the golden ratio.
Any positive integer can be represented as a sum of powers of $\phi$ with integer exponents:
This representation is called the phinary representation of an integer.
For an integer n, we define M(n) as the sum of all exponents in its phinary representation: $$ M(641) = \sum{641_{\textrm{base}\phi}} = \sum{\{13,9,7,5,2,0,3,11,14\}} = 8$$ In the range $\left[1,10000000\right]$, find the minimun for M, the first integer which reaches the minimum, the maximum, and the first integer which reaches the maximum
Answer format: $M_{min},n_{min},M_{max},n_{max}$ // 17,77,8,46 for the range $\left[1,100\right]$
[My timing: 10 sec]
P.S:
Thanks to sinan for his suggestions which made this problem more interesting.
Any positive integer can be represented as a sum of powers of $\phi$ with integer exponents:
 $ 1= \phi^0$
 $ 2= \phi^1+\phi^{2}$
 $ 3= \phi^2+\phi^{2}$
 $ 4= \phi^2+\phi^0+\phi^{2}$
 $ 5= \phi^3+\phi^{1}+\phi^{4}$
 $ 6= \phi^3+\phi^1+\phi^{4}$
 $ 7= \phi^4+\phi^{4}$
 $ 8= \phi^4+\phi^0+\phi^{4}$
 $ 9= \phi^4+\phi^1+\phi^{2}+\phi^{4}$
 $10= \phi^4+\phi^2+\phi^{2}+\phi^{4}$
This representation is called the phinary representation of an integer.
For an integer n, we define M(n) as the sum of all exponents in its phinary representation: $$ M(641) = \sum{641_{\textrm{base}\phi}} = \sum{\{13,9,7,5,2,0,3,11,14\}} = 8$$ In the range $\left[1,10000000\right]$, find the minimun for M, the first integer which reaches the minimum, the maximum, and the first integer which reaches the maximum
Answer format: $M_{min},n_{min},M_{max},n_{max}$ // 17,77,8,46 for the range $\left[1,100\right]$
[My timing: 10 sec]
P.S:
Thanks to sinan for his suggestions which made this problem more interesting.
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