The world is simply full of numbers of all kinds. All the clutter on numbers was made by none but man. Fibonacci and Lucas numbers; perfect numbers; Fermat numbers; Multiply-perfect numbers; abundant and super-abundant numbers, practical and impractical (!) numbers, etc. However, there is an interesting correlation between all these junk of numbers. In this entry, I try to let us get an insight into Perfect, Multiply-perfect, Abundant and Super-abundant and Practical numbers.
As usual, all of us have realized from the definition of a perfect number that what a multiply-perfect number is.
Definition and properties of the function
Let us, for the sake of clarity, define our notations first. We denote as the sum of the divisors of , and as the number of the divisors of . We try to express in terms of the prime divisors of . Note that, by the fundamental theorem we have that any can be expressed in the form of (where ‘s are the primes arranged in order or not in order)
A simple combinatorial argument leads to the obvious and well-known fact that
(including as a factor).
Now, for a formula of we see that all the divisors are of the form
and . Thence the sum, can be factored into (or rather, found out to be) terms in the expansion of
Now it is evident that when we have the multiplicity property of this function, ie . The (obvious) proof is left to the reader.
Before discussing the general multiply-perfect numbers, we dive into the perfect numbers for a little time. Let us try to list some of the perfect numbers. They are written in order. The next values can be generated with a computer, but what do we see common in these numbers?
(1) (Obviously) All are even and ; which makes them perfect.
(2) and is a prime.
and is a prime.
and is a prime.
and is a prime.
Proceeding in this manner, Euler proved (again, how many theorems did he prove?) that if is a given number with the property that is a prime, that is going to be a perfect number.
With this (extremely) brief idea of perfect numbers we are able to give a generalization. We consider a number such that its where is some natural number.
Before getting some information and problems on multiply-perfect numbers, let us try a problem.
If is a perfect number and are primes, find all possible values of
Obviously considering modulo we have that for some . If so, we consider In this case,
, which is way too bizarre for a perfect number . So . So we get . Indeed, and and are all primes.
Let us denote a multiply perfect number of the order by whereas . We play around with a few multiply-perfect numbers of some orders. Just as in the previous case, we try to list some properties of a number. At the first glance into a tuply perfect number like or (Verify with a calculator or the prime factorization formulae for ), we might try to conjecture a few things. Like, all are even. We might try to construct a sequence of these numbers for different , and conjecture that there does or does not exist such a sequence. These conjectures are needed in mathematics, but we still need a proof of those, that is why they are called conjectures.
Very few has been known about large ‘s, so let us start by noting some examples.
Let be a number such that does not divide yet it satisfies , show that O, ie is a perfect number of order .
It is obvious from the multiplicative property of that
so that is an type number.
Every number such that has at least different prime factors.
This, also is obvious on using the prime decomposition of , which is
Now using which is obvious on expanding, we have
Now with the same old technique, we consider the product upto terms, which is surely going to be (because the statement is modified to contradict this event by origin); therefore we are surely done.
A number is known as superabundant if and only if exceeds for all .
The sequence of numbers does not have any upper bound..
If are the divisors of then we have, (why?) and summing up on this sequence we get as,
Since we have to prove that this has no upper bound, let for our convenience. Then we have,
Then we have where is the harmonic series. As we know the harmonic series on natural numbers does not have any upper bound as increases, and therefore is unbounded from above.
The numbers with the property that are abundant ones.
are abundant at our first instance, and we can also find out that all abundant numbers are even.
We have a property regarding the abundant numbers.
Every number is abundant implies that is abundant,
Solution to be found by the reader.
We will discuss on some Practical, Highly Composite Numbers and Quasi-perfect, Semi-perfect numbers soon.
Hope you do not get insane with a lot of numbers, we will get some exercises on these too.
(To Be Updated Soon)