By

Joseph L. Pe

iDEN System Engineering Tools and Statistics

Motorola

21440 West Lake Cook Road

Deer Park, IL 60010

Email: ajp070@email.mot.com

**1.
****Summary**

This paper presents a notion of perfect numbers relative to arithmetical functions:

an arithmetical function f produces a set of *f-perfect numbers*. Two among the many examples considered
are small “perturbations” of the normal definition; late in these two
sequences, odd perfect numbers appear! (Could the situation be similar for the
usual perfect numbers?) Also, this paper generalizes amicable pairs and
sociable chains. Mysteries and open problems abound for those who like a
challenge. There is also *Mathematica *code to start the reader on his own
explorations.

**2. ****Preliminaries**

In this discussion, *number *means
natural number. Functions are called *arithmetical *if their domains are
sets of integers. For simplicity, an arithmetical function in this paper will
also be integer-valued and include the set of natural numbers in its domain.

**3. Perfection is in the eye of
the beholder**

The
following generalization considers perfect numbers relative to an arithmetical
function f. Under this scheme, the usual perfect numbers

6, 28, 496, 8128, 33550336, 8589869056
....

appear
as one among many species of “f-perfect numbers”.

Let
n have proper divisors d_{1}, ..., d_{k}. (A *proper divisor*
of n is a divisor of n other than n itself.) Recall that n is *perfect* if
n is the sum of its proper divisors, that is,

which
can be written as

where
id is the identity function defined by id(n) = n. This suggests, for a
generalization, replacing id by an arbitrary arithmetical function f.

For
an arithmetical function f, the *f-perfect numbers* are the numbers n such
that

where
d_{1}, ..., d_{k} are the proper divisors of n.

Hence,
the ordinary perfect numbers are id-perfect numbers, where id is the identity function.

**4.
Two canonical perfect number sequences**

The
sequence of G-perfect numbers with G(n) = n + 1 begins

4, 10, 44, 2336, 8896, 34432, 449295,
....

and
the first g-perfect numbers with g(n) = n - 1 are

12, 196, 368, 1696, 30848, 437745....

These
appear in the Online Encyclopedia of Integer Sequences (http://www.research.att.com/~njas/sequences/Seis.html)
as A066229
and A066230,
respectively. Whenever a sequence in this paper appears in the OEIS, its OEIS
number appears with it.

For
example, consider the G-perfect number 44. The proper divisors of 44 are 1, 2,
4, 11, 22. G(44) = 45 = 23 + 12 + 5 + 3
+ 2 = G(22) + G(11) + G(4) + G(2) + G(1), verifying that 44 is G-perfect.

Here
is Mathematica code to generate the G-perfect numbers.

Obviously,
it can be modified for other functions to yield other perfect number sequences.

**5.
In search of a pattern**

The
(ordinary) even perfect numbers can be generated by Euclid’s expression 2^{n-1}(2^{n}
- 1), with prime n making (2^{n} - 1) a prime. Can one find a similar
expression for, say, the G-perfect numbers? Euclid’s formula was probably
discovered by looking for patterns in the prime factorizations of the perfect
numbers. For example, the following Mathematica code

to
get the prime factorizations of the first seven perfect numbers yields

where
the factorization of, say, 496 can be read as 2^{4} x 31^{1 }(from{{2,
4},{31, 1}} in the first line). It is readily observed that the perfect numbers
are products of powers of 2 and primes
3, 7, 31, 127, 8191,…. In fact, doing a quick search for this prime sequence on
the OEIS identifies it as the sequence of Mersenne primes.

A similar experiment on the G-perfect
numbers yields

Although
several G-perfect numbers follow the pattern of a power of 2 times a prime, the
first one is a power of 2, and the seventh one does not even have 2 as a prime
factor. Also, the first six numbers have at most two prime factors, but the
seventh one has five. The only property these numbers seem to have in common is
that every prime factor > 2 has multiplicity 1, but even this could be
disproved by some larger G-perfect number.

Comparing
the means of, say, the first four (n + k)-perfect numbers for k = -3, -2, -1, 0, 1, 2, 3 reveals only more
irregular behavior, as in the following list.

__ __

__k__** Mean
of first four (n+k)-perfect numbers**

3
60.25

2
2533.00

1 598.50

0 2164.50

-1 568.00

-2
37.50

-3 53637.50

**6.
Odd perfect numbers and other open problems**

What
is certain from these considerations is that an expression generating G-perfect
numbers is far from obvious. One finds a similar situation with the g-perfect
numbers. The reader is invited to tackle these apparently difficult open
problems.

However,
not all is chaos and darkness: the last numbers appearing in the above G- and
g-perfect number lists turn out to be odd. Finally, after initial runs of even
perfect numbers, odd perfect numbers appear (even if they are “only” G- or
g-perfect)! Could this reflect a similar state of affairs for the usual perfect
numbers? At the time of writing, all known perfect numbers are even, but
perhaps there is actually an odd one waiting out there in the mysterious realm
of the very large….

**7.
More examples**

Obviously,
other functions f besides the simple ones mentioned above can be used. If f
grows too fast, say f(n) = n^{2}, then no perfect numbers are obtained.
At the other extreme, if f grows too slowly, it can generate “too many” perfect
numbers. For example, it is not hard to see that the φ-perfect numbers are
the powers of 2, and the d-perfect numbers are the squares of primes.
(φ(n) denotes Euler’s totient function, giving the number of numbers
relatively prime to n and less than n; d(n) denotes the number of divisors of
n.) The **1**-perfect numbers, where **1** is the constant function
mapping each number to 1, are the prime numbers. (Any nonzero number besides 1
can be used.) The **0**-perfect numbers consist of all numbers > 1.

However,
there is a great variety of interesting f. Of course, the choice of which f are
interesting is somewhat personal. Nevertheless, most people will agree that
there should be relatively few f-perfect numbers. Preferably, the first few
f-perfect numbers should share some striking property or pattern, inviting a
general conjecture. Or, as in the case of the G- and g-perfect numbers, they
should resist simple classification, thereby challenging one to find
closed-form generators.

Here
are a few more examples. Again, each of these sequences invites a search for a
closed-form generator.

Recall
that σ(n) denotes the sum of the divisors of n.
The sequence of σ-perfect numbers begins

198, 608, 11322, 20826, 56608,
3055150.....

(
A066218).
For example, the proper divisors of 198 are 1, 2, 3, 6, 9, 11, 18, 22, 33, 66,
99; and the sum of their σ-values = 1 + 3 + 4 + 12 + 13 + 12 + 39 +
36 + 48 + 144 + 156 = 468 = σ(198). Hence, 198 is σ-perfect. Does an odd σ-perfect
number exist?

Let
h(n) = σ(φ(n)). The sequence of h-perfect
numbers begins

2, 88, 328, 5128, 9075, ....

(
A066226).
Again, an odd perfect number appears relatively late in the sequence.

The
sequence of j-perfect numbers with j(n) = φ(σ(φ(n)))
begins

2, 4, 28, 40, 448, ....

(
A066228).
There are no more terms less than 10^{5}. Are there any more terms? Is
this an infinite sequence?

f
can even be an arithmetical function defined from a transcendental one.
Consider k(n) = Floor(|n sin(n)|) (that is, the greatest integer __<__ |n
sin(n)|). The first few k-perfect numbers are

3, 6, 10, 34, 50, 91, 222, 364, 1485,
6640, ....

(A066245).

Of
course, these examples only skim the surface of the ocean of “interesting”
perfect number sequences. Perhaps the reader will have fun experimenting with
his own functions and discover something intriguing and beautiful in the
process.

**8.
Generalized amicable numbers and sociable chains**

f-amicable
numbers* *are defined similarly.
Specifically, let D(n) be defined by

where
d_{1}, ..., d_{k} are the proper divisors of n. Then a, b form
an *f-amicable pair* if f(b) =
D(a) and f(a) = D(b).

It is always
delightful to find some striking relationship between apparently unrelated
numbers. The discoverer of the amicable pair 220, 284 must have been very
pleased. I have had the satisfaction of uncovering three amicable pairs for
g(n) = n - 1 and two pairs for G(n) = n + 1. Here are the g-amicable pairs:

100, 110; 1806, 2402; 1872, 3742.

(A066511).

For example,
the proper divisors of 100 are 1, 2, 4, 5, 10, 20, 25, 50. g applied to these
divisors yields 0, 1, 3, 4, 9, 19, 24, 49, whose sum is 109. So D(100) =
g(110). The proper divisors of 110 are 1, 2, 5, 10, 11, 22, 55. g applied to
these divisors yields 0, 1, 4, 9, 10, 21, 54, whose sum is 99. So D(110) =
g(100). Therefore, 100, 110 form a g-amicable pair.

The G-amicable pairs are

36, 62; 168, 326.

(A066505).

There
is only one σ-amicable pair whose members do not
exceed 4000: 312, 584. One can modify the above Mathematica code for perfect
numbers to deal with amicable pairs. However, exhaustive search for amicable
pairs rapidly becomes computationally infeasible.

Can
the reader add to these lists?

In
general, one can search for *f-sociable number chains*. For example, a *crowd*
is a sociable chain of length 3. It is not known whether crowds exist (cf.
[W]). Correspondingly, an *f-crowd *is a triple of numbers a, b, c with
D(a) = f(b), D(b) = f(c), D(c) = f(a). Can the reader find an f-crowd for an
“interesting” function f?

[A] Andrews, G. “Number Theory”, Dover Publications, NY 1994.

[B] Beiler, A. “Recreations in the Theory of Numbers”, Dover Publications,

NY 1964.

[S]
Sloane, N. *The Online Encyclopedia of Integer Sequences *at the following

URL: http://www.research.att.com/~njas/sequences/Seis.html

[W]
Wells, D. “The Penguin Dictionary of Curious and Interesting Numbers”,

Penguin Books,
1998.