A generalization of a theorem by Cheo and Yien concerning digital sums
Internat. J. Math. & Math. Sci.
Vol.
A GENERALIZATION OF A THEOREM BY CHEO AND YIEN CONCERNING DIGITAL SUMS
CURTIS N. COOPER 0
ROBERT E. KENNEDY 0
0 Department of Mathematics and Computer Science Central Missouri State Univerqity Warrensburg , Missouri 64093 U.S.A
For a non-negative integer n, let s(n) denote the digital sum of n. Cheo and Yien proved that for a positive integer x, the sum of the terms of the sequence (1.5) [m]o where the constant implicit in the big-oh notation is dependent on k. The following notation will be used to facilitate the proof of (1.2). For integers x and y,
-
(1.1)
(1.2)
(1 4)
is (4.5)xlogx + O(x)
the sum of the sequence
is also (4.5)xlogx + O(x)
on k.
{s(n)
0, I, 2
(x-l)}
{s(kn)
0, I, 2
(x-l)}
KEY WORDS AND PHRASES. Digital sums.
1980 MATHEMATICS SUBJECT CLASSIFTCATION CODE.
10H25
INTRODUCTION.
In Cheo and Yien [1]
it was proven that for a positive integer x
In this paper we let k be a positive integer and determine that
The constant implicit in the big-oh notation is dependent
xxs(n)
(4.5)xlogx + 0(x)
(4.5)xlogx + 0(x)
[m]j
m mod 10j
where s(n) denotes the digital sum of n. Here, we will show that, in fact, for any
positive integer k,
x rood y (I .3)
will be the remainder when x is divided by y and, as usual, square brackets will denote
the integral part operator. In addition, for non-negative integers m, i, and j we let
[m]Ji [[m]JJ i
for i j.
Thus, the j right-most digits of m are given by (1.4) and the number determined by
dropping the i right-most digits of m is given by (1.5). Therefore, the number
determined from the jth right-most digit of m to the (i + l)st right-most digit of m
is given by (1.6).
2. A PROOF OF (1.2) WHEN k AND i0 ARE RELATIVE PRIME.
Let (k,10) i, x be a positive integer, and L [logx] Then
xn=0
s(kn)
xn=0
xs([kn]L) +
xn=O
s([kn]L)
and
an so
Then,
[x] L+l-i i0L+l-i
x
[
x .
n=O
n 0
+
+
THEOREM BY CHEO AND YIEN CONCERNING DIGITAL SUMS
for each i, it follows that
[x -L+2-ilL+l_i 10L+l-i
for each i.
Now since
n 0
by [
2
], we have that
lOL+l-iL
xi
S ([kn] e+l-i)
L+2-i.
([X]e+l_i
10L+l-i
n
0
s(n)
s(n)
4.5(L +
i)10L+l-i
s([kn] u+l-i)
i n xi_
Using (2.16) and (2.11) in (2.8), by (2.2) we have the expression given in (1.2). The
constant implicit in the big-oh notation is dependent on k with k and I0 relatively
prime.
3. CONCLUSION.
For any positive integer k, there exists non-negative integers a, b, and r such that
k 2a5br with (r,10) i. Note that if k r, then we have (1.2). However, by use of
the following generalization to Lemma 2, and some technical modifications, it can be
shown that the restriction that k and I0 be relatively prime can be removed in the
x-.
derivation of (2.1). That is,
s(kn) (4.5)xlogx + 0(x) (3.1)
n 0
for any positive integer k.
LEMMA 3. Let k 2a5br with (r,lO)
and i
max {a,b}. Then for any
nonnegative integer d,
(4.5)xlogx + 0(x)
(3.2)
(3.3)
I. CHEO, P. and YIEN, S. A Problem on the K-adic Representation of Positive Integers,
Acta Math. Sinica 5 (1955), 433-438.
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2. KENNEDY , R.E. and COOPER , C.N. On the Natural Density of the Niven Numbers , College Math. Journal I_5 ( 1984 ), 309 - 312 .
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