Deficient number

In number theory, a deficient number is a number in which the sum of its proper factors is less than the number.

For example, the number 4 is deficient.

The proper factors of 4 are 1, and 2. Since 1 + 2 = 3 and 3 is smaller than 4, 4 is  deficient.

Notice that 4 is not included among the proper factors of 4.

15 is also deficient.

The proper factors of 15 are 1, 3, and 5. Since 1 + 3 + 5 = 9 and 9 is smaller than 15, 15 is deficient.

42 is not a deficient number.

The proper factors of 42  are  1, 2, 3, 6, 7, 14, 21, 42

Since 1 + 2 + 3 + 6 + 7 + 14 + 21 = 54 and 54 is bigger than 42, 42 is not a deficient number.

The sum of the proper factors of the number must be smaller than the number, not greater.

The first 20 deficient numbers are 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, and 25.

From the list above, you can see that all the prime numbers from 1 to 25 are included in the list. Having said that we can say with confidence that all prime numbers are also deficient numbers.

The reason that all prime numbers are deficient numbers as well is because the sum of all proper factors of a prime number is always going to be 1.




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