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.