Table of Contents

- 1 How do you determine if a number is abundant or deficient?
- 2 Is 60 abundant or deficient?
- 3 Is 48 abundant deficient or perfect?
- 4 What is the first deficient number?
- 5 Is 32 abundant deficient or perfect?
- 6 What is special about the number 64?
- 7 When do you call a number abundant or deficient?
- 8 Is there an infinite number of abundant numbers?

## How do you determine if a number is abundant or deficient?

This leads to the Greek classification of numbers as follows:

- If P(n) > n, then n is called an abundant number.
- If P(n) < n, then n is called a deficient number.
- If P(n) = n, then n is called a perfect number.

## Is 60 abundant or deficient?

The first 28 abundant numbers are: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, (sequence A005101 in the OEIS). For example, the proper divisors of 24 are 1, 2, 3, 4, 6, 8, and 12, whose sum is 36.

**What makes a number deficient and abundant?**

Every number can be classified as abundant, deficient, or perfect, according to the following definitions: Abundant: The sum of the proper factors is greater than the number itself. Deficient: The sum of the proper factors is less than the number itself.

**What are the divisors of 64?**

Divisors of numbers

Number | Prime factorization | Divisors |
---|---|---|

61 | 61 | 1,61 |

62 | 2*31 | 1,2,31,62 |

63 | 63 | 1,63 |

64 | 25 | 1,2,4,8,16,32,64 |

### Is 48 abundant deficient or perfect?

Is 48 abundant deficient or perfect?

N | Divisors of N | Notes |
---|---|---|

47 | 1, 47 | Deficient |

48 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 | Abundant |

49 | 1, 7, 49 | Deficient |

50 | 1, 2, 5, 10, 25, 50 | Deficient |

### What is the first deficient number?

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).

**What are the twin primes between 1 to 100?**

The first few twin prime pairs are: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), … OEIS: A077800. for some natural number n; that is, the number between the two primes is a multiple of 6.

**Is 17 abundant deficient or perfect?**

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.

#### Is 32 abundant deficient or perfect?

Is 32 abundant or deficient?

N | Divisors of N | Notes |
---|---|---|

29 | 1, 29 | Deficient |

30 | 1, 2, 3, 5, 6, 10, 15, 30 | Abundant |

31 | 1, 31 | Deficient |

32 | 1, 2, 4, 8, 16, 32 | Deficient |

#### What is special about the number 64?

It is the smallest number with exactly seven divisors. It is the lowest positive power of two that is adjacent to neither a Mersenne prime nor a Fermat prime. 64 is the sum of Euler’s totient function for the first fourteen integers. 64 is also the first whole number that is both a perfect square and a perfect cube.

**What is the factors of 64?**

Factors of 64

- Factors of 64: 1, 2, 4, 8, 16, 32 and 64.
- Negative Factors of 64: -1, -2, -4, -8, -16, -32 and -64.
- Prime Factors of 64: 2.
- Prime Factorization of 64: 2 × 2 × 2 × 2 × 2 × 2 = 26
- Sum of Factors of 64: 127.

**Why is 1 a deficient number?**

In order for a number to be a deficient number, the sum of the proper factors of the number must be smaller than the number, not greater, or equal to the number. 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.

## When do you call a number abundant or deficient?

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.

## Is there an infinite number of abundant numbers?

Every multiple of an abundant number is itself abundant, so there is an infinite number of abundant numbers. In 1998, the mathematician Marc Deleglise showed that roughly one-quarter of all the positive integers are abundant. Deficient numbers occur more frequently than abundant numbers.

**What did Nicomachus mean by abundant and deficient?**

Later, the philosopher Nicomachus,* in his Introduction to Arithmetic, would coin the terms “abundant” and “deficient,” attaching moral qualities to these numbers.

**Which is the most abundant number in the world?**

Abundant Numbers. Twelve is the first abundant number. The next abundant number is 18 because the proper divisors sum to 21 (1 + 2 + 3 + 6 + 9). The first five abundant numbers are 12, 18, 20, 24, and 30.