What is a deficient number?

What is a deficient number?

In number theory, a deficient number or defective number is a number n for which the sum of divisors of n is less than 2n. For example, the proper divisors of 8 are 1, 2, and 4, and their sum is less than 8, so 8 is deficient.

How are deficient numbers calculated?

A number is called deficient if the sum of its divisors is less than the number itself. For example, if we find and add all the divisors of the number 10 we get 1 + 2 + 5 = 8. Comparing them, we see that 8 is less than 10, therefore 10 is a deficient number.

What does deficient mean in math?

Learn about this topic in these articles: …than the number; in a deficient number, the sum of its proper divisors is less than the number. For example, 24 is abundant, its divisors giving a sum of 36; 32 is deficient, giving a sum…

Is 22 a deficient number?

. 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.

Is 31 a deficient number?

Deficient Numbers Any positive integer that exceeds the sum of its distinct proper factors. Any prime number is deficient, because it has only one proper factor: 1. All numbers of the form 2n are also deficient. Example: 32 (=25) is a deficient number because the sum of its distinct proper factors is 31 (1+2+4+8+16).

Is 36 a deficient number?

So, 36 is not a deficient number!

Is 1000 an abundant number?

J. Broscius (around 1652) showed that there are only 21 abundant numbers between 10 and 100 and all of them are even; the only odd abundant number less than 1000 is 945.

Is 1 a deficient number?

Primes, prime powers, and any divisors of a perfect or deficient number are all deficient. The first few deficient numbers are 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, (OEIS A005100).

Is 19 a deficient number?

Deficient numbers occur more frequently than abundant numbers. In other words, the sum of the proper divisors of most numbers is less than the numbers themselves. Examples of deficient numbers include 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, and 23.

How many deficient numbers are there between 1 and 20000?

There are 15043 deficient, 4 perfect, 4953 abundant numbers between 1 and 20000. Three algorithms presented, the first is fast, but can be a memory hog when tabulating to larger limits. The second is slower, but doesn’t have any memory issue. The third is quite a bit slower, but the code may be easier to follow.

What are abundant and deficient numbers?

As an extension of the idea of perfect numbers, the concept of “abundant” and “deficient” numbers emerged. If the sum of the proper divisors of a number is greater than the number itself, then the number is called abundant or excessive. The proper divisors of 12 are 1, 2, 3, 4, and 6.

What are some examples of deficient and perfect numbers?

Examples of deficient numbers include 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, and 23. Perfect numbers, which occur infrequently, are the most interesting of the three types.

What are defective numbers like 8?

Numbers like 8, whose proper divisors have a sum that is less than the number itself, are called deficient or defective. The Greeks, who regarded the proper divisors of a number to be the number’s “parts,” were the first to refer to perfect numbers — numbers that are the exact sum of their parts.