Aurora Saturday 14:30 Python Practice 22.06.25.

Starting with x and repeatedly multiplying by x, we can compute x³¹ with thirty multiplications:

x² = x × x, x³ = x² × x, x⁴ = x³ × x, …, x³¹ = x³⁰ × x.

The operation of squaring can be appreciably shorten the sequence of multiplications. The following is a way to compute x³¹ with eight multiplications:

x² = x × x, x³ = x² × x, x⁶ = x³ × x³, x⁷ = x⁶ × x, x¹⁴ = x⁷ × x⁷, x¹⁵ = x¹⁴ × x, x³⁰ = x¹⁵ × x¹⁵, x³¹ = x³⁰ × x.

This is not the shortest sequence of multiplications to compute x³¹. There are many ways with only seven multiplications. The following is one of them:

x² = x × x, x⁴ = x² × x², x⁸ = x⁴ × x⁴, x⁸ = x⁴ × x⁴, x¹⁰ = x⁸ × x², x²⁰ = x¹⁰ × x¹⁰, x³⁰ = x²⁰ × x¹⁰, x³¹ = x³⁰ × x.

If division is also available, we can find a even shorter sequence of operations. It is possible to compute x³¹ with six operations (five multiplications and one division):

x² = x × x, x⁴ = x² × x², x⁸ = x⁴ × x⁴, x¹⁶ = x⁸ × x⁸, x³² = x¹⁶ × x¹⁶, x³¹ = x³² ÷ x.

This is one of the most efficient ways to compute x³¹ if a division is as fast as a multiplication.

Your mission is to write a program to find the least number of operations to compute xⁿ by multiplication and division starting with x for the given positive integer n. Products and quotients appearing in the sequence should be x to a positive integer’s power. In others words, x⁻³, for example, should never appear.

The input is a sequence of one or more lines each containing a single integer n. n is positive and less than or equal to 1000. The end of the input is indicated by a zero.

Your program should print the least total number of multiplications and divisions required to compute xⁿ starting with x for the integer n. The numbers should be written each in a separate line without any superfluous characters such as leading or trailing spaces.