THE ON NATURAL DENSITY OF RANGE

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805
Internat. J. Math. & Math. Sci.
VOL. 12 NO. 4 (1989) 805-808
ON THE NATURAL DENSITY OF THE RANGE
OF THE TERMINATING NINES FUNCTION
ROBERT K. KENNEDY and CURTIS N. COOPER
Departmeit of Mathemat[cs and Computer Science
Central Missouri State University
Warrensburg, Missouri 64093
and
VLADIMIR DROBOT and FRED HICKLING
Departmellt of Mathematics
Santa Clara Unlversity
Santa Clara, CA 95053
(Received January 15, 1988 and in revised form June 6, 1988)
ABSTRACT.
Noting that the expression
tl
[_n__]
t
gives the number of terminating nines
tO
which occur up to n but not including n, we will denote the above expression by t(n)
call t the "terminating nines function".
T= it(n): n=1,2,3, ...} will be determined.
The
and
KEY WORDS AND PHRASES.
density
of
the
set
Digital sums, terminating nlnes, natural density.
1980 AMS SUBJECT CLASSIFICATION CODES.
I.
natural
Primary IIA63, Secondary: none.
INTRODUCTION.
The number
A(x).
of
positive
d(A)
integers
in
d(A), of the
The natural density,
llm
X-
provided this limit exists.
a
set
A,
not
exceeding x,
is
denoted by
set A is defined as
A(x)
X
The determination of the natural density of a given set
of positive integers is an important topic in most number theory textbooks and is the
subject
o
much research.
For example, the set of positive integers
N
n:
s(n) is a factor of n},
where s(n) denotes the digital sum of n, is the set of Niven numbers [I] and was shown
to have
a natural denslty of 0
digital sum function.
in
[2].
Here, we are interested in a part of the
R. K. KENNEDY, C. N. COOPER, V. DROBOT AND F. HICKLING
806
It has been shown that
s(a)
n- 9
tl
[-!*--I
t
[0
where, as usual, the square brackets denote the integral part operator.
Noting that
the express ton
[_n_]
t
10
t
g[es the number of terminating nines which occur up to n but not including n, we will
the "ter,nlnatLng nines function".
denote (l.l) by t(n) and call
1,2,3, ...}
[t(rO:n
of the set T
,Note that T
T.
does not include every positive integer since, for example, 10
2.
The natural density
will be determined [n what follows.
NOTATION AND TERMINOLOGY.
In what follows, we will say that the terminating nines function, t, has a "jump"
of size k at an integer a [f t(a)--t(a-l) + k.
ends with exactly k nines.
only if a
Thus, t has a jump of size k
[f and
To determine the natural density of T, we
first show that
T(t(n)
t(n)
n/
9
10’
where T(t(n)) is the number of members of T not exceeding t(n).
To do this, we will
}.
is the
t(n)
If
count how many integers are missing from set {t(1), t(2),...,
number of these missing integers, then it follows that
T(t(n))
t(n)
THE NATURAL DENSITY OF T.
Noting that if
n and t has a jump of size k at a,
a
produce k-I missing integers.
at a for
a
<
Moreover, each missing integer
Thus, each
n.
a
< n, such that I0 k dvldes
divide a, produces k-I missing integers.
O’s in all integers
0.
a
=
]-
jl
we have that
T(t(n))
n
[-].
n
[-i-6],
a but I0
k+l
Jump
does not
is the number of terminating
n, minus the number of integers
Therefore, since
n
Hence, a n
then this jump will
is a result of some
a
n which end with
NATURAL DENSITY OF THE RANGE OF THE TERMINATING NINES FUNCTION
807
Using the above, we thus co,clude that
[]-]
r(t(n)
t(n)
n
[]-o-1
02_
+
+...
which may be written as
n
-Fd +
T(t(n))
t(n)
n___ +
I0
since the denominator is equal
n
to
]- + 0(1).
Thus,
o(t)
+
AlO-
+ O(log n)
n
--+ --+
2
n
+ O(log n), and the numerator is equal
[0
i0
n
T(t(n))
t(n)
lira
n
1-- +
llm
n
n
--+
13
0(I)
+
__n__
10-
+ 0(log n)
9
Letting x be an arbitrary integer, and y be such that
t(y) < x
we have that
t(y)
<
t(y + 1),
t(y) does
O(log x) since x
not exceed the number of digits
in x.
Since, T(x)
T(t(y)), we have
T(t(y)__)
T(x)
x
T(t(y)
t(y) + O(log x)
x
and so, by the above limit, It follows that
T(x)
x
-
9
10
Stating this as a theorem we have:
THEOREM I.
function.
4.
Then
Let T
d(T)
t(n): n
1,2,...
where
t
is
the terminating nines
9
GENERALIZATION TO BASE b.
Finally, it should be noted that the development given above and Theorem
generalized to any integral base bo
if
tb(n)
in the base b representation of the sequence of positive integers up to n,
have the following generalization of Theorem I:
THEOREM I’.
Let
{tb(n):
n-- 1,2
}.
can be
denotes the number of terminating b-l’s
Then d(T)
b-I
--o--
then we
808
R. K. KENNEDY, C. N. COOPER, V. DROBOT AND F. HICKLING
REFERENCES
I.
KENNEDY, R., GOODMAN, T. and BEST, C. Mathefaatical
Numbers, The MArYC Journal 14 (1980), 21-25.
2.
KENNEDY, R. and COOPER, C.
College Math..Journal
Discovery and Niven
On the natural density of the Niven numbers,
I_5 (1984), 309-312.
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