Fourth Equation!

This is the last question! Great job to those who have done all the previous ones! (will add links below) 

Again this question was made by UIntOverflow , Thankyou for the help!  

Michael is going to the bank to deposit $65,000. He tells the banker, "I will be back next week to withdraw some money anywhere from $1 to $65,000, but I will be an extreme hurry so I cannot wait for you to count the cash. You need to have ready some envelopes with prefixed amount of cash to match my withdraw amount." The banker agrees, and is able to come up with a brilliant system of distributing cash into multiple envelopes so that no matter what Michael asked for, the banker can give Michael a set of envelopes that is equal to the withdraw amount. What is the minimum number of envelopes needed to achieve this goal?

Example: If Michael asked for $100, the banker has options of giving him two $50 envelopes, one $100 envelope, or even 100 $1 envelopes. 

Good luck! 

Here's your joke: 🙄

Descartes walks into a bar and the bartender asks "Would you like a drink?" Descartes replies "I think not" and promptly disappears. 

Ok, so after a day or so I will publish an additional cipher so that you can translate your letters. 8)

Please keep answers to yourself so as to allow others to solve it for themselves!  

Numbers to letters cipher: https://www.chess.com/blog/Max_Wolfe/i-hate-negative-numbers-ill-stop-at-nothing-to-avoid-them

1st equation: https://www.chess.com/blog/Max_Wolfe/first-equation

2nd equation: https://www.chess.com/blog/Max_Wolfe/second-equation

3rd equation: https://www.chess.com/blog/Max_Wolfe/third-equation