537
Internat. J. Math. & Math. Scl.
Vol. 8 No. 3 (1985) 537-543
ON AN ASYMPTOTIC FORMULA FOR THE NIVEN NUMBERS
CURTIS N. COOPER and ROBERT E. KENNEDY
Depar%zent of .ath,:atics and Computer Science
Central Missour State University
Warrensburg, Missouri
64093 U.S.A.
(Received May 20, 1985)
s a positive integer which is d.visible by its digital
of an asymptotic formula for N(x) is given.
possibiS.ity
um. A
A partial result
numbers less than x
of
Niven
Here, N(x) denotes the nmber
which
will be preented. This result wll be an asymptotic formula for
denotes the number of Niven numbers less than x with digital sum k
ABSTRACT.
A Niven number
d.sc.,ss.on of the
Nk(X)
WOPDS AND PHRASES. Digital sums, asymptotic formula, Niven number.
0A30
0H25
9S0 ATHEMATICS SUBJECT CLASSIFICATION CODE.
},Y
I.
I[FROD[!CT!ON.
[]
the concept of a Niven number was introduced as any
One of the
positive integer n which is divisible by its digital sum s(n)
of Niven numbers which was investigated was the
first questions about the set, N
In Kennedy et al
status of
x
-
L N(x)
x
(.)
(In what follows,
denotes the number of Niven numbers less than x
A(x) will be
then
of
set
integers,
a
is
we will use the convention that if A
the number of members of A less than x .) This limit, if it exists, is called
where
N(x)
the "natural density" of the set
N
(the natural densit?
Even %hough this was answered in Kenned and Cooper [2]
of N is zero), other questions demanded attention. In particular, "Can an
asymptotic formula for N(x) be determined?" That is, does there exist a function
f(x)
such that
Lira
x--
If such an
?
lx)
(.2)
exists, then the uual notation to indicate this is,
f(x)
N(x)- (x)
The following notation will be used to arrive at a partial, answer to this
qe’t,",n.
I.et
k
s positive ’nteger.
Then
aS
k
may be written in the form
.)
538
C. N. COOPER AND R. E. KENNEDY
(t, 0)
where
We define:
s(x)=
Sk--
k}
and
o=
.(k)
o uo t.
of
(.s)
Nk(X
In what. follows, we will develop an asymptotic formula for
c
A, ASY,:PTCIC FORMULA WHEN k
?.
2asb
Nk(X
S,h a fo,u!a for
or
0
c
when
2
emma.
I..E:Y: I. Tet
can easily
found for
This Is given in
Theorem 2.1
k
of the fo
2a503c
with the help of the
fo! owing
+ege.
be
n
Then
e,
He, the
%he pare,’nse ,enot,
PRO’?,
Nk(10 n)
atest
brackets denote the
--i,are
(2.!.)
integer function and
;ote that an intec,r of the fom.,
bnom.a7 coeff$.e-ien%.
f
+
ie(k)
i=O
where
e
i
{0, I]
c
0
’/ C
+
02
+
+
Cf
k
and
(2.)
is a Niven number with digita] sum
But the sequence
k
f
(2.)
can be rearranged exactly
(2.)
ways, and each of these will determine a Niven mmber with digital sum k
T.h,:refore, we have that
Tt" 2. "* t:.
(2,1) hods.
_.
wh.?",
.;..() ,-,,
PROOF.
Let
,
c
,,
or
(o)!:
":e the positive integer such that
10
n
x
<
I0 n+l
Then
539
ASYITOTIC FORILA FOP. THE NIVEN NUNBERS
Sk(10n)’In mlNnk/k!k
k
we have
By Lemma
not dependent on n
(k)
Sk(iOn
Nk(lOn
and so,
Nk(lOn)
since each side of
(2.10)
is asymptotic
Nk(lO n)
nk/k:
to nk/k!
(2.)
From
Nk(lOn+1)
Nk(X)
(2.2)
and
(n,+l)k/k! , Nk(i.On+l
Nk(10n).# nk/k:
it readily follows that
() ,,,
.
.o,=e
(o
)/:
[ogx] o-
n
A LOW 0 0 ,()
not. he that Lemma I can be used to de%eine a lower
!t should
In fact, the seah for such a lower dl to the meth
bound for N(x)
To deteine is lower
that will give us an asptotic foula for
m for some positive integer m
m
and (k)
e(k)
Then
bound, let k 2
"
Nk(X
and we have by Lemma i
that
N2m(lon)
2m
Thus,
N(IOn)
eventually.
N2m(lOn)
That is, there exists an integer
N(IOn)
for all
n
n
K
K such that
n 2m-I
Note that the rightmost inequality of
n of degree
establish that eventually
is a polynomial in
2m_l
2
N(x)
m
(3.2)
follows since
As in the proof of Theorem 2
>
(og .) 2m-
we can then
C. N. COOPER AND R. E. KENNEDY
540
Therefore, for any integer t
m
for any integer
t
eventual ly.
AT ,3"MTCTC FO:0iI,A FOR N
In what follows, for positive integers
a.
k(x)
[0,
x
decimal represer.tation of
no.-oegatve negers less than
iO
iO
n
n
and
k
we will denote the
n
[0,
where
.I_0
n)
is the: set of
as
n-i
i
xi10
i=O
Note that initial zeros will be allowed so that x will have n digits. For
e(k)- we also define the finite sequences B(x,j) and
2,
each j
0,
_,
T(x)
x
(4.2)
i
i--O
and
f
B(x,j)
x
(k)+
j +
ie(k
0
i
whre
f
(4.2)
Using
x
=
y
and
(.3), we
T(x)
if and only if
[n- (k)e(k) 13
(4.4)
=
nowdeflne the relation
on
0n) by:
T(y) and B(x,j) is a reamngement of the
trms of B(y,j) for each j
It is clear that
is an equivalence latlon on
memr
of
[0,
=
0n)
.
let
The fo].owlng ].ea wil
I.E’W.
POOP.
j=o
x,. y
let
Z!:,oe
i=o
x
=
y
<x
denote the
us
to help count the
0 n)
we hav tha%
Then
[0
I0 n)
For x a
ulvalence cass contalni x
x
T(x)
j=o
nr
=
of Nien
v .:e:3....
T(y)
and
i=o
nm
that
)
s(r,
"s a
with
.,
(v
reae-
ASYMPTOTIC FORMULA FOR THE NIVEN NUMBERS
541
(.8)
m
(.9)
of digits, let
for
t
set.
O, I, 2,
Hee, the # symbol denotes the oadinality of the
is the number of tems of the sequenoe
then d
9
For example, if t
3
Therefore, the number of finite sequences whleh cn be formed by
rearranging the terms of (4.9) is given by the multinomial coefficient
equal to 3
0’
dl’ ""’ d9
We will. use this fact to develop an asymptotic formula for N
[0, los)
Nk
5. Let x
to
or
equal
less
than
of degree
I,EMNA
Then
.#<x>
k(x)
for any integer
is a polynomial in
f
where
k
e(Ik
PROOF.
B(x,j)
Note that each
for various
dt(J)
j’s
<x>
y
Let
#{0
i
f
may be found by rearranging the terms of
X(k)
t}
+ j + le(k)
(4.13)
By the previous discussion, the number of such y’s which can be formed by these
rearrangements ,s given by
_e(k)-l_
jl= I
/
o(j), d1(j), ...,
d9(Jd
f .+
’
(4.1h.)
542
C. N. COOPER AND R. E. KENNEDY
But each factor of this product is a pol3momial in
of degree
d()
i=I
is a polynomial in
f
j=O
of degree
i=l
Tess than or equal to k
THEOEEM 6. Let n, k be positive integers.
which is
Nk(lOn
for a constant
PROOF.
c
cn
Then
k
which depends upon
Since
X#<x>
Nk(10n)
where the sum is +.aken over the collection of equivalence classes induced
on
EO, !0n)
N
k
not exceeding
Nk(10n
we have that
is a polynomial in
f
(4.’q)
by :
of degree
k by Lemma 5
Thus, all we need to do in ozder to show that
has degree k is to constmct a Niven number, x, with digital sum k
Such an x is
is a polynomlal in f of degree k
such that
Nk(!On
x>
k-
I
10(k)
+ ie(k)
(4.19)
i=O
Here,
(a..2o)
which is a polynomlal in
f
of degree exactly
k
Th us,
Nk(iOn) clfk
which is dependent on
for some
k
Since
(422)
we have that
f
n/e()
and therefore,
Nk(lOn)
cn
k
wher
c
k-cl/(e(k))
Finally, us.ng a.n argument similiar to that in the proof of Theorem 2
have the followng corollary
COROLLARY 7. Let k, x be positive integers. Then
Nk(x
where
c
depends on
k
c(log x) k
we
(.23)
543
ASYMPTOTIC FORMULA FOR THE NIVEN NUMBERS
5.
CONCLUSION.
Thus, a partial answer concerning an asymptotic formula for N(x) has been
presented. As was shown by Theorem 2 exact values of the constant c can be
it is indeed
In fact, given a partic,]ar k
found for caln integers k
would
involve an
This
will
c
that
be.
form
exact
the
oossible to determine
_nvetigatin. of the partitions of k with summands less than or equal to 9
and the number o solutions to certain d.ophantlne congruences. We feel that
+.his is a subject for future stu4y. Tb. determination of an asymptotic formula
or ..,,T(x)
.
.
however, wi]. be ]e/ an an open problem.
,,,
.
7-OCD,’.,.T, T.A. BST, C,H. Mathematical Discovery and Nven
[..,NNEDY,
(I 980), 2-25
T.h,,be, The MATYC Jou:ra
,’.NEDv R.
N O the atu Density oY the Niven Numbs,
.OOPE
TG, S..
o
An Improvement ad Generalza%[on of Bellman-Shaper’s Thom
n dw Nu Th,o,
ez. ah.
l (963),
Po].
9-2o.
roc.
s..