Create binary sequence to use less inputs
6 pulldown resistors, one to each data line. 1 to 5 diodes from each button, the button connected to vcc. 

Button 1 has one diode to pull up the d0 line when pressed.

Button 61 has 5 diodes to the d5, d3, d2, d1, d0 lines as for your example above. 

Button 0 not used. When any button is pressed, at least one of the data lines will go up. Can be used for interrupt on button press.

Suitable pulldowns, almost any value, 10k will do.
Suitable diodes, almost any diode, 1N4148 will do, quite cheap by the 100s.

This is a static solution, simple, but uses shed-loads of diodes. There are others. 

You could put them into a matrix, scan and decode.
You could put them into a shift register.

As Arduino has ADC inputs, you could use each button to switch various value resistors, to change the voltage into the ADCs. Whether you could get enough resolution to resolve all 61 buttons without calibration or adjustment is another matter, you might want to split it into 4 channels of 16 buttons.