Internat. J. Math. & Math. Sci.
VOL. 11 NO. 2 (1988) 343-350
343
SOME RESULTS CONCERNING EXPONENTIAL DIVISORS
PETER HAGIS, JR.
Mathematics Department
Temple University
Philadelphia, PA 19122
Received November 18, 1986 and in revised form February 9, 1987)
a
ABSTRACT.
If the natural number n has the canonical form
bl b2
d
br
e-perfect number if
(e)
m+n
(m)
o
hi+
(e)
o
(e)
(ni)-n i.
o(e)(n)
(n); no,n
a
2
Pl P2 "’’Pr
is said to be an exponential divisor of n if
Pl P2 "’’Pr
The sum of the exponential divisors of n is denoted by
o
a
o(e)(n),
bila i
r
then
2
for i
r.
n is said to be an
2n; (m;n) is said to be an e-amicable pair if
is said to be an e-aliquot sequence if
2
Among the results established in this paper are: the density
n
of the e-perfect numbers is .0087; each of the first 10,000,000 e-aliquot sequences
is bounded.
KEYS WORDS AND PHRASES.
e-aliquot sequences.
Exponential divisors, e-perfect numbers, e-amicable numbers,
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
10A20.
INTRODUCTION.
If n is a positive integer greater than one whose prime-power decomposition is
given by
a
n
a
Pl P2
2
a
Pr
r
(I.I)
b
then d is said to be an "exponential divisor" of n if d
1,2
for i
o(e)(n).
r.
b
b
2
Pl P2 "’’Pr
r
where b
i
la i
The sum of all of the exponential divisors of n is denoted by
This function was first studied by Subbarao [I] who also initiated the
study of exponentially perfect (or e-perfect) numbers.
The positive integer n is said to be an e-perfect number if
o(e)(n)"
o(e)(n)
2n.
If
kn, where k is an integer which exceeds 2, n is said to be an e-multi-
perfect number.
The properties of e-perfect and e-multiperfect numbers have been
investigated by Straus and Subbarao [2] and Fabrykowski and Subbarao [3].
It has
been proved, for example, that all e-perfect and e-multiperfect numbers are even.
Also, if n is an e-perfect number and
3n
then
21101
n and n >
10618
While it is easy to show that there are an infinite number of e-perfect numbers,
whether or not any e-multiperfect numbers exist is still an open question.
Subbarao,
Hardy and Aiello [4] have conjectured that there are no e-multiperfect numbers.
have
proved that any which exist are very large.
They
344
P. HAGIS
In Section 2 of the present paper the density of the set of e-perfect numbers is
Section 3 is devoted to a study of e-amicable pairs, integers m and n
investigated.
(e)
such that o
(e)
ni+
2.
m+n
(m)
(ni)-n.1
o
for i
(e)
(n).
Finally, e-aliquot sequences
where
nO,nl,n 2
are studied in Section 4.
0,1,2
THE DENSITY OF THE e-PERFECT NUMBERS.
By definition,
o(e)(1)
o(e )(p)
Therefore, since
and it is easy to see that
p if p is a prime, we see that o
o(e)(n)"
(e)
(m)
is multiplicative.
m if m is square-
free.
Now suppose that n, as given by (I.I), is a powerful e-perfect number (so that
ao1
2 for i
then o
1,2
(e)(mn)
r and o
(e)(n)
2n).
Then if (m,n)
ns
<
<
are the powerful e-perfect numbers which do
2
E(x), the set of (all) e-perfect numbers less than or equal to x,
positive number and n
not exceed x then
and m is squarefree
Therefore, if x is a (fixed)
2mn so that mn is an e-perfect number.
< n
s
E(x)
is given by
A
A.1 where
U
i=l
{mni:(m,n i)
i
I, m
=<
x/n.l
and m is
squarefree}
Let N be a positive integer and let X be a positive real number.
(2.1)
If Q(N,X) is
the number of positive, squarefree integers which do not exceed X and which are
relatively prime to N, then E. Cohen (Lemma 5.2 in [5]) has shown that
8(N)’X + 0(e(N)’X
Q(N,X)
((2)
where 8(N)
(l+I/p))
I/2)
-I and 8(N)
(2.2)
is the number of squarefree divisors of
pl m
N.
It is easy to see that 8(N)
2.
(k) is the Riemann Zeta function, so that
pl
(2)
2/6,
and the constant implied by the O-term is independent of N and X.
If Q(e,x) is the number of e-perfect numbers which do not exceed x (so that
Q(e,x) is the cardinality of E(x)) it follows from (2.1) and (2.2) that
s
s
Q(e x)
x
i=l
0(xl/2 [
B(ni)/ni
+
8(n i)/n
+ 0(x -I/2
8
i=l
1/2 ).
(ni/n i
Therefore,
s
Q(e,x)/x
[.
i=l
i
s
[
i=l
8(n i)/n I/2 ).
(2.3)
i
The following results concerning powerful numbers will be needed in what follows.
Proofs may be found in Golomb [6].
LEMMA I.
If r
< r
2
<
is
thesequenceofpowerful numbers then
I/r i
i=l
is convergent.
LEMMA 2. If P(X) is the number of powerful numbers not exceeding x then
P(x) < 2.2x 1/2 for large x.
Now let e be a given positive number and let
Pi
denote the ith prime.
There
exists a positive integer k such that
2/P k
e
(2.2K)
-I
/3
where K is the constant implied by the 0-term in (2.3).
(2.4)
345
SOME RESULTS CONCERNING EXPONENTIAL DIVISORS
Since there are only a finite number of powerful
e-perfect numbers which are
divisible by fewer that k distinct primes (see Theorem 2.3 in [2]) there exists a
<
is the sequence of powerful e-perfect
2
numbers then for all i > J n. has at least k distinct prime factors and n i has a
l
I/2
Hp where the
Since
prime factor, say Qi’ such that Qi
ni is powerful, n i
(2.4)
from
follows
it
and
of
n
factors
over
distinct
prime
the
product is taken
< n
positive integer J such that if n
Pk"
i,
J then
that if i
pn
8(ni)/ni
2/p
<
2/P k <
2/Qi-<-
.
i
e’(2.2K)-I/3.
(2.5)
J (with J held fixed) we can
Splitting the sum in the O-term in (2.3) at i
J
I/2 < e/3
-I/2
At the same time, since
-Ktake x large enough so that x
i
i=l
every n is powerful, we see from (2.5) and Lemma 2 that we can also take x large
8(ni)/n
i
enough so that
x
s
-1/2
"
K"
112
8(ni )/nI/21" x<
i=J+l
x
i=l
_
-I/2
s
(2.2K)-I/3
Ki=J+l
P(x)"
e"
(2.2)
-I
/3
<
Finally, since 8(n i) <
and every n i is powerful we see from Lemma
8(ni)/ni
(This series
is convergent.
m_
/3.
that
be finite since whether or not the set
of powerful e-perfect numbers is finite or infinite is an open question).
It follows
that we can take x (and consequently s) large enough so that the tall of this series
is less that
e/3.
Therefore, from (2.3wehave for all large values of x,
IQ(e,x)/x
8(n
i=l
i)/nil
(2.6)
< e
We have proved
Let Q(e,x) denote the number of e-perfect numbers which do not
be the sequence of powerful numbers. Then
exceed x and let n
n < n <
3
2
THEOREM I.
lira
where 8(n)
6w
[
Q(e,x)/x
x+
-2
i=l
(l+I/p)
8(ni)/n i
-I
C
Correct to ten decimal places
C
0086941940
I010:
36; 1800; 2700;
(There are eight powerful e-perfect numbers less than
value of C
The
357,739,200.
approximate
17,424; 1,306,800; 4,769,856; 238,492,800;
given above was calculated using these eight numbers).
agrees
The "theoretical" density of the e-perfect numbers as given in Theorem
very nicely with the following exact computational results:
7
.0086940;
.008690;
Q(e,106)/106
3.
Q(e,107)/lO
Q(e,108)108
Q(e,105)/105
.00871;
.00869417.
EXPONENTIALLY AMICABLE NUMBERS.
We shall say that m and n are exponentially amicable (or e-amlcable) numbers if
o
(e)(m)
m
+
n
o
(e) (n).
(3 I)
346
P. HAGIS
LEMMA 3.
If (m;n) is an e-amicable pair and p is a prime, then
plm
if and
only if
PROOF.
o
(e)
pa Im where
Suppose that
zs a multiplicative function.
same argument, if
pln
COROLLARY 3.1.
a
I.
plo (e)(m)
Then
since
plo (e)(pa) and
pln. By the
It is now obvious from (3.1) that
then
If (m;n) is an e-amicable pair then mn(mod 2).
If (m;n) is an e-amicable pair and there is no prime p such that pllm and
we shall say that m and n are primitive e-amicable numbers. It is easy to see that
if (m;n) is a primitive e-amicable pair and r is a squarefree positive integer such
that (m,r)
i, then (rm;rn) is an amicable pair.
A search was made for all primitive e-amicable pairs (m;n) such that m < n and
m
The search required about 1.5 hours on the CDC CYBER 750 and three pairs
were found. They are as follows:
107.
(22327
(2332527
192", 22337219); (22327
19 2333527219).
Are there any odd e-amicable
This list suggests the following questions.
numbers?
Is every e-amicable number
Are there any powerful e-amicable numbers?
divisible by at least four distinct primes?
612; 22347261);
(It is easy to show that every e-amicable
number has at least three different prime factors).
The following result can sometimes be used to generate new e-amicable pairs from
known pairs.
Suppose that (aM;aN) is an e-amicable pair such that (a,M)--(a,N)= I.
(e) (a)/a
then (bM,bN) is an e-amicable pair.
and o
(e) (aM) o
o(e) (bM) o (e) (b) o (e) (M) a-lb( (e) (a) o (e) (M)
bo
THEOREM 2.
If (b,M)
PROOF.
a
4
-1
-<-
b (aM
o(e)(b)/b
(b,N)
+ aN)
a-
bM
+ bN
Similarly, o
(e)(bN)
bM
+
bN
The results of a computer search for powerful numbers a and b such that
a < b < I0000 and
are given in Table I.
o(e)(a)/a o(e)(b)/b
TABLE I
(e)(a)/a
3/2
22
413
32
2232
39/32
2352
3352
233552
2332
12/7
2272
or
223352
2752
26
5/3
2411
or
2233
or
233352
235272
65/48
2732
2633
40/21
233272
22337,2
347
SOME RESULTS CONCERNING EXPONENTIAL DIVISORS
Since (2
EXAMPLE.
since
4
(2
4.
(e)(2 2)/2
II
2
3
2
2
2
3
(e)
o
2
19-2;
2
7
2
7
2
4
)/ 2
112
4
19
4
3
2
2
2
3
12
3
7
3
7
2
19) is an e-amicable pair and
it follows from Theorem 2 that
2
19) is an e-amicable pair.
EXPONENTIAL ALIQUOT SEQUENCES.
s(e)
The function
is defined by s
(e) (0)
o
s(e)(r)
s(e)(1)
a.liquo.t divisors of n.
(e) (n)
(e)
(n)
n, the sum of the
exponential
0 for every squarefree number r and we
A t-tuple of distinct natural numbers (no;n
nt_ I) with
(e)
t-I and s
(nt_ I) n o is called an exponential
ni s(e) (ni_ I) for i 1,2
t-cycle. An exponential l-cycle is an e-perfect number and an exponential 2-cycle
define s
0.
A search was made for all exponential t-cycles with smallest
is an e-amicable pair.
member not exceeding 10
7
None with t > 2 was found
The exponential aliquot sequence (or e-aliquot
defined by n
(e)
s
n,n
O
s(e) (hi_ I)
(no) ni
sequence){ n i
with leader n is
Such a sequence is said to
0 for i > k). An
is squarefree for some index k (so that n i
k
exponential aliquot sequence is said to be periodic if there is an index k such that
be terminating if n
(nk;nk+l;...
nk+t_ I)
An e-aliquot sequence which is
is an exponential t-cycle.
neither terminating nor periodic is unbounded.
107.
About
An investigation was made of all aliquot sequences with leader n -<
and
be
to
terminating
found
were
9,896,235
2.3 hours of computer time was required.
103,765 were periodic (103,694 ended in l-cycles and 71 ended in 2-cycles).
The fact that the first ten million exponential aliquot sequences are bounded
might tempt one to conjecture that the set of unbounded e-aliquot sequences is empty.
However, the following theorem shows that e-aliquot sequences exist which contain
arbitrarily long strings of monotonically increasing terms.
Therefore, whether or
not unbounded e-aliquot sequences exist would seem to be a very open and difficult
question.
Then there exist
Let N be a positive integer which exceeds 2.
THEOREM 3.
< n
infinitely many exponential aliquot sequences such that n o < n
PROOF.
2
qil
Let ql,q2,...,
+ I) for
qN
be a sequence of N primes such that
<
<
2
ql
2,
nN_ 2.
q2
3 and
(Infinitely many such sequences exist since, by
(qi+l
2
contains an infinite number
Dirichlet’s theorem, the arithmetic progression aqi
2
K qi"
of primes.) We shall write qi+l +
no
Now let
2 2
ql q2
N-I.
2,3
i
i
be the exponential aliquot sequence with leader n o given by
n0,nl,n2,..,
2
qN" Then
N
(e)(n0)
i=l
+
(qi
N
q2i)
3
N-I
3
qlq2
qN
qN
qlq2
i--2
(I + qi
2
Kiql
i=l
and
nl
o(e) (n o
-n
o
(3
qlq2
qN
K1
-I-
qN2)
N-I
Therefore, n
M
ql where
(Ml,qi)
for i
1,2
N-I.
N-I
i=l
2
qi
348
P. HAGIS
n0/36 is
o(e) (n0/36)
72
n
n0/36
> 72
o
=
nk
and
n
(e)
o
k
Therefore, n
o
<
<
n
(e)(36)
o
(e)
(n0/36)
n
o
N-2
(e)
(e) (36)
N-k
1,2
for i
(nk-1 /36)
nk-1
nk_
nk_
nN-2
36_ 2 where (6,_ 2) I.
(e)
o
(_2) 36%_ 2 > 72_ 2 36%_ 2
REMARK I.
o
o
o
(,qi)
nk-1
n.K-I /36
< n
o
2
qi where
(nk--1)
> 72
n
n
for k-- 2,3
Similarly, we find tha
N-k
(e)(n o
o
not squarefree, n
Since
If
nN_ 2
is not squarefree, then
-2
36_ 2 --nN_ 2.
The proof of Theorem 3 is modeled on that of Theorem 2.1 in [7].
(e) (n)/n), the mean value of o [e)"
Our next objective is to determine M(o
-I Ng
f(n)
lim N
The mean value of an arithmetic function f is defined by M(f)
n-N+
We shall need the following lemma due to van der Corput (See Theorem A in [8] .)
72
nN_
REMARK 2.
"(n)/n.
LEMMA 4.
If f and h are arithmetic functions such that f(n)
h(d) and
d |n
[.
h(n)/n is absolutely convergent then M(f)
n=l
h(n)/n.
(e)
o
(n)/n.
n--i
We wish to apply this lemma to the function f(n)
By the Moebius
(e)(d)/d.
h is multiplicative and h(1)
(n/d)o
dn|
If p is a prime and a is a positive integer then h(p a)
inversion formula, h(n)
I.
=o(e)(pa)/pa-o(e)(pa-l)/p a-l.
If a < 6 it is easy to verify that
-I
-2 <
<
p
p
.)
lh(p3)
lh(P a)
Since
p-3/4
p-
o(e) (pa)/pa
o(e)(pm)/pm
"’hpa)[
<
<
[.
< p
-a/4
(For example,
Suppose that a
o(e) (pa-1)/pa-1
or
->
6.
p/(p-l)p (a-l)/2.
Since a
Ih(p a)
o(e)(pb)/pb
6 it follows easily that
(e)
-lh(pa)
< p -el4
h(n)/n is absolutely convergent so that Lemma 4 applies if f(n)
n--I
.
(pa)/pa.
l,
-I/4 for every positive
n
integer n.
lh(n)
-
->
Then
(e) (pa-)/pa-
+ p/(p-1)pm/2 (see [21 or [41)and
Since h is multiplicative,
that
lh(pa)
It follows
o(e)(n)/n.
From Theorem 286 in [9] we have
n=l
h(n)/n
{I + h(p)/p + h(p
/p2
+ ...}
p
T-[ {l+p-l(o (e) (p) ip_l) +p-2(o(e)(p2)Ip2-o(e)(p)Ip) +...}
P
p
[ o(e)(pj)ip2j
p-I [
=0
{(I
o
(e) (pj)/p2j
j =0
p-l)
p
o(e)(pj)/p2j}.
j=O
Now the last infinite series can be "split up" by first taking all the terms
with numerator
pJ
to form the series
I pj/p2j
j --0
I i/pJ;
=0
then taking the remaining
.
349
SOME RESULTS CONCERNING EXPONENTIAL DIVISORS
the terms with numerators p
2
.
j=2
to form the series
taking the terms with numerators p
etc.
p/p 2j
[
terms with numerators p to form the series
3
p-3
p2/p4j
.
0=2
to form the series
(p-2 )J; then
(p-4)
p-6
.
j=0
=0
p
3
p6j
p
-9
taking
then
p-6
--0
=2
It follows that
{(I
h(n)/n
n=l
p-I )((1
p
-I
p-9 (I
-1)-1
+
p-2) -1
p-3(1
p
p-4
+ p -6 (I
=
{(i
p
-I
+
)((I
p
p-6) -I
-I -i
+
+ ...)}
(p3
p)
-I
p2
+ (p 6
-I
P
+
(p9_ p3)-I
=]--["
+ ...)}
{1 + (1- p
-I
>’.
(p 3j
pj
-I
j=l
p
From Lemma 4 we have
THEOREM 4.
M(o
(e)(n)/n)
{I + (I
p
-I
(p 3j
pj
-I
C.
j=l
p
1.136571.
Correct to 6 decimal places, C
(This approximate value of C was calculated using all primes less than
106
in
the infinite product.)
Since
s(e)(n)"
COROLLARY 4.1.
Finally, since
o[e)(n)" n we
M(s(e)(n)/n)"
ni+i/n i
have
.136571.
s(e)(ni)/ni
we see that, in some sense, the average
value of the ratio of two consecutive non-zero terms of an e-aliquot sequence is
about .136571.
REFERENCES
I.
SUBBARAO, M.V. On some arithmetic convolutions, The Theory of Arithmetic Functions, Lecture Notes in Mathematics 251 (1972), Springer-Verlag, New York,
247-271.
2.
STRAUS, E.G. and SUBBARAO, M.V.
465-471.
3.
FABRYKOWSKI, J. and SUBBARAO, M.V. On e-perfect numbers and a conjecture of
Straus and Subbarao, Congressus Numerantium 52 (1986), 79-90.
4.
SUBBARAO, M.V., HARDY, G.E. and AIELLO, W. On the existence of e-multiperfect
numbers, The Fibonacci Quarterly 25 (1987), 65-71.
5.
COHEN, E.
6.
GOLOMB, S.W.
7.
te RIELE, H.J.J.
8.
COHEN, E. ARITHMATICAL NOTES, I.
12 (1961), 214-217.
9.
HARDY, G.H. and WRIGHT, E.M.
On exponential divisors, Duke Math. J. 41 (1974),
Arithmetical functions associated with the unitary divisors of an
integer, Math. Zeit. 74 (1960), 66-80.
Powerful Numbers, Amer. Math. Monthly
77
(1970), 848-855.
A theoretical and computational study of generalized aliquot
sequences, Mathematical Centre Tracts 74 (1976), Amsterdam.
On a theorem of van der Corput, Proc. A.M.S.
An Introduction to the Theory of Numbers (Fourth
Edition) Oxford University Press, New York, 1960.