Search a number
A number    is called admirable if there exists a divisor    of    such that  . In other words,    is equal to the sum of its proper divisors, where one of them has minus sign.

For example, 12 is admirable because its proper divisors are 1, 2, 3, 4, 6, and 12 = 1 + 2 + 3 - 4 + 6.

Clearly, admirable numbers are a subset of abundant numbers and they are infinite because, for example, all the numbers  , with    prime, are admirable.

The largest number that cannot be written as a sum of admirable numbers is 1003.

Pairs of consecutive admirable numbers are rarer than pairs of consecutive abundant numbers. Up to    there are only two such pairs, namely (29691198404, 29691198405) and (478012798575, 478012798576).

Below, the spiral pattern of admirable numbers up to  . See the page on prime numbers for an explanation and links to similar pictures.

The smallest 3 × 3 magic square made of admirable numbers is

 282 678 102 174 354 534 606 30 426

The first admirable numbers are 12, 20, 24, 30, 40, 42, 54, 56, 66, 70, 78, 84, 88, 102, 104, 114, 120, 138, 140, 174, 186, 222 more terms

Admirable numbers can also be... (you may click on names or numbers and on + to get more values)