Let
denote the sum of the distinct divisors of an integer
(cf.
Divisor;
Number of divisors).
The integer
is called
abundant
if
;
deficient
if
;
and
perfect
if
(cf. also
Perfect number).
Note that some authors call a number
abundant if
.
Clearly, these numbers are in fact perfect or abundant (i.e.
"non-deficient" )
numbers.
In
[a5],
L.E. Dickson
gives details on the early history of abundant numbers.
G. Nicomachus
(about
100)
separated the even numbers into
abundant,
deficient and perfect, and dwelled on the ethical
importance of the three types.
A.M.S. Boethius
(around
500),
in a Latin exposition of the
arithmetic
of Nicomachus, stated that perfect numbers are rare, while
abundant
( "superfluous" )
and deficient
( "diminutos" )
numbers
are found to an unlimited extent.
N. Jordanus
(around
1236)
stated that every multiple of a perfect
or abundant number is abundant.
He attempted to prove the erroneous statement that all abundant
numbers are even.
C. Bovillus
(around
1509)
corrected this statement, by citing
and its multiples.
Bachet de Méziriac
(around
1600)
gave a proof that
is perfect if
is a
prime number,
and abundant if
is composite. He remarked that the odd number
is abundant.
J. Broscius
(around
1652)
showed that there are only
abundant
numbers
between
and
and all of them are even;
the only odd abundant number less than
is
.
(The statement by
E. Lucas
(1891)
that
is the smallest odd abundant number is probably a misprint for
.)
Ch. de Neuveglise
(1700)
proved that the products
of two consecutive numbers are abundant,
and all multiplies of
or an abundant number are abundant.
J. Struve
(1808)
considered abundant numbers which are products
of three distinct prime numbers in ascending order; for
,
,
or
,
and for
,
,
,
is abundant for any prime number
.
Of the numbers
,
are abundant.
Dickson
(1913,
[a6])
called a non-deficient number
primitive abundant
if it is not a multiple of a smaller
non-deficient number.
He proved that there are only a finite number of primitive
non-deficient numbers having a given number of distinct odd
prime
factors and a given number of factors
.
There is no odd abundant number with fewer than three distinct
prime factors, the primitive ones with three are
He gave also a table of all even abundant numbers
.
Dickson's result was a starting point for much further
research.
In
1949
and
1968,
H.N. Shapiro
(
[a20],
[a21])
proved the following result. Let
be a rational number.
A necessary and sufficient condition that there exist infinitely
many
primitive
-abundant numbers
(i.e.
but
for all
,
)
with
distinct prime factors is that
has a representation
with
,
,
where
.
Here,
is the Euler
totient function
and
denotes the number of distinct prime factors of
.
In
1933,
F. Behrend,
H. Davenport
and
S. Chowla
[a4]
showed that
the density of non-deficient numbers exists and is positive.
This result follows also from a
theorem
of
P. Erdős
[a7]
stating that the sum of reciprocals of primitive abundant
numbers converges.
Let
be the counting function of primitive
-abundant
numbers.
Erdős proved that
[a10]
and that
[a8]
This was sharpened successively by
A. Ivić
[a13],
with
in place of
and
in place of
;
and by
M.R. Avidon
[a2],
who considered
in place of
,
and
in place of
.
L. Alaoglu
and Erdős
[a1]
call a number
superabundant
if
for all
.
Let
be the counting function of superabundant numbers. For
two consecutive superabundant numbers
,
they prove that
and this was sharpened to
for an infinity of
by
J.-L. Nicolas
[a16].
Alaoglu and Erdős showed that
,
while Erdős and Nicolas
[a11]
demonstrated that
.
Alaoglu and Erdős
[a1]
introduced also the notion of
highly abundant number,
a number
with the property that
for all
.
For the counting function
of these numbers one has
for all
and large
;
if
is highly abundant, then the largest prime factor of
is less than
.
Erdős and Nicolas
[a11]
call a number
cube-free superabundant
if
implies
,
where
for
and
for
(with
a prime number and
a
positive integer). They prove that if
and
are two consecutive cube-free superabundant numbers, then
.
A non-deficient number is called
weird
by
S.J. Benkovski
and Erdős
[a3]
if it is not pseudo-perfect (cf. also
Perfect number).
They proved that the density of weird numbers
is positive.
V. Siva Rama Prasad
and
D.R. Reddy
[a23]
say that a number
is
primitive unitary
-abundant
if
but
for all
,
(
).
Here,
denotes the sum of unitary divisors of
(for these functions, as well as related results, see also
[a15]).
Let
be the set of these numbers. Then
Miscellaneous results.
Let
.
A number
is called
-non-deficient
if
.
By sharpening a result of
O. Grün
[a12],
H. Salié
[a18]
proved that the least prime factor of every
-non-deficient
number with
prime factors is less than
.
Ch.R. Wall
[a24]
proved that there exist infinitely many abundant integers
(with
and
given). Let
be fixed. Then there exist
consecutive abundant numbers.
There exist infinitely many sequences of five consecutive
deficient numbers.
(See
[a25].)
See
[a14]
for a table of odd primitive abundant numbers
with five distinct prime factors for which
If
,
the number
is abundant, see
[a22].
For others results on deficient, perfect, or related numbers,
see
[a15],
[a8],
[a9],
[a19],
[a17].
L. Moser
[a26]
proved that every integer
can be expressed as the sum of two abundant numbers. Actually, this is valid for
integers
,
see
[a27].
For a table of abundant numbers less than
,
see
[a28].