Deficient number

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A Deficient number is a number $n$ for which the sum of $n$'s proper factors is less than $n$. For example, 22 is deficient because its proper factors sum to 14 < 22. The smallest deficient numbers are 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, and 17.

Problems

Introductory

Problem 1

Prove that all prime numbers are deficient.

Solution

Problem 2

Prove that all powers of prime numbers are deficient.

Solution

See Also

Perfect number

Abundant number

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