Sunday, April 4, 2021

Abundant number, Perfect number and Aliquot parts - Definition, Example, Usage

 

Abundant number, Perfect number and Aliquot parts - Definition, Example, Usage,pseudo perfect number

Abundant number, Perfect number and Aliquot parts - Definition, Example, Usage

Abundant number

A number that is smaller than the sum of its aliquot parts (proper divisors). Twelve is the smallest abundant number; the sum of its aliquot parts is 1 + 2 + 3 + 4 + 6 = 16, followed by 18, 20, 24, and 30. A weird number is an abundant number that is not semi perfect; in other words, n is weird if the sum of its divisors is greater than n, but n is not equal to the sum of any subset of its divisors. The first few weird numbers are 70, 836, 4,030, 5,830, and 7,192. It isn’t known if there are any odd weird numbers. A deficient number is one that is greater than the sum of its aliquot parts. The first few deficient numbers are 1, 2, 3, 4, 5, 8, and 9. Any divisor of a deficient (or perfect) number is deficient. A number that is not abundant or deficient is known as a perfect number.

Perfect number

A whole number that is equal to the sum of all its factors except itself. For example, 6 is a perfect number because its factors, 1, 2, and 3 add to give 6. The next smallest is 28 (the sum of 1 + 2 + 4 + 7 + 14). Augustine (354–430) argued that God could have created the world in an instant but chose to do it in a perfect number of days, 6.

Early Jewish commentators saw the perfection of the universe in the Moon’s period of 28 days. The next in line are 496, 8,128, and 33,550,336. As René Descartes pointed out: “Perfect numbers like perfect men are very rare.”

All end in six or eight, though what seems to be an alternating pattern of sixes and eights for the first few perfect numbers doesn’t continue. All are of the form 2n – 1 (2n− 1), where 2n− 1 is a Mersenne prime, so that the search for perfect numbers is the search for Mersenne primes. The largest one found is 23021376 (23021377 − 1). It isn’t known if there are infinitely many perfect numbers or if there are any odd perfect numbers.

A pseudo perfect number or semi-perfect number is a number equal to the sum of some of its divisors, for example, 12 = 2 + 4 + 6, 20 = 1 + 4 + 5 + 10.

An irreducible semi-perfect number is a semi-perfect number, none of whose factors is semi-perfect, for example, 104.

A quasi-perfect number would be a number n whose divisors (excluding itself) sum to n + 1, but it isn’t known if such a number exists.

A multiply perfect number is a number n whose divisors sum to a multiple of n. An example is 120, whose divisors (including itself) sum to 360 = 3 × 120. If the divisors sum to 3n, n is called multiply perfect of order 3, or tri-perfect. Ordinary perfect numbers are multiply perfect of order 2. Multiply perfect numbers are known of order up to 8. See also abundant number.

Aliquot part

Also known as a proper divisor, any divisor of a number that isn’t equal to the number itself. For instance, the aliquot parts of 12 are 1, 2, 3, 4, and 6. The word comes from the Latin ali (“other”) and quot (“how many”).

An aliquot sequence is formed by taking the sum of the aliquot parts of a number, adding them to form a new number, then repeating this process on the next number and so on. For example, starting with 20, we get 1 + 2 + 4 + 5 + 10 = 22, then 1 + 2 + 11 = 14, then 1 + 2 + 7 = 10, then 1 + 2 + 5 = 8, then 1 + 2 + 4 = 7, then 1, after which the sequence doesn’t change. For some numbers, the result loops back immediately to the original number; in such cases the two numbers are called amicable numbers. In other cases, where a sequence repeats a pattern after more than one step, the result is known as an aliquot cycle or a sociable chain.

An example of this is the sequence 12496, 14288, 15472, 14536, 14264, . . . The aliquot parts of 14264 add to give 12496, so that the whole cycle begins again.

Do all aliquot sequences end either in 1 or in an aliquot cycle (of which amicable numbers are a special case)? In 1888, the Belgian mathematician Eugène Catalan (1814–1894) conjectured that they do, but this remains an open question.

 

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