Neurobics

Almost Perfect Numbers

Russell Dear

The so-called "perfect numbers" fascinated the early Greeks. They are those numbers which equal the sum of their divisors, including unity but excluding themselves. 28 is a perfect number since 1+2+4+7+14 = 28. Perfect numbers are rather thin on the ground -- the first five are 6, 28, 496, 8128 and 33550336. The early Greeks knew of the first four but the next was not discovered until the middle ages. In 1985 the 30th perfect number was determined by Chevron Geosciences on their CRAY supercomputer -- who knows what's happened since then.

Various offspring have emerged over the years and you might like to have a go at finding some of them. To ease the descriptions, in each of the following the d-divisors of a number are defined to be all the numbers which divide into it excluding the number itself. Thus the d-divisors of 10 are 1, 2 and 5.

Almost Perfect numbers

Their d-divisors sum to one less than themselves, for example 8 is almost perfect since 1+2+4 = 7. The first three almost perfect numbers are 2, 4 and 8. Find the fourth. It is not known whether an odd almost perfect number exists.

Quasi-Perfect numbers

Their d-divisors sum to one more than themselves. It is not known if any exist but if one does it must be the square of an odd number.

Semi-Perfect numbers

Numbers which are the sum of some of their d-divisors. The first three are 12, 20 and 24. Find the fourth.

Multiply-perfect numbers

The d-divisors of such numbers sum to a multiple of themselves. Perfect numbers are, of course, also multiply-perfect. Such a number is 672 since its d-divisors sum to 1344 = 2 x 672. What is the smallest multiply-perfect number which is not perfect?

Abundant numbers

Numbers which are less than the sum of their d-divisors. The first three are 12, 18 and 20. Find the fourth.

Weird numbers

Numbers which are abundant but not semi-perfect. That is, weird numbers are less than the sum of their d-divisors but not the sum of any set of their d-divisors. They are rare. 836 is one but there is another much smaller, see if you can find it.

Deficient numbers

Numbers which are more than the sum of their d-divisors. Most numbers are deficient.

Amicable numbers

Pairs of numbers which are each the sum of the d-divisors of the other. The smallest such pair is 220 and 284. Pythagoras is said to have known of this pair but no others. Perhaps that's not surprising as the next smallest pair is 17296 and 18416.

Untouchable numbers

Numbers which are never the sum of the d-divisors of any other number. The two smallest untouchable numbers are 2 and 5. What is the next smallest?

Russell Dear is a Mathematician living in Invercargill