A generalization of a theorem by Cheo and Yien concerning digital sums

International Journal of Mathematics and Mathematical Sciences, Jul 2018

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{s(n):n=0,1,2,…,(x−1)}is (4.5)xlogx

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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. Advances in ns Research Hindawi Publishing Corporation ht p:/ www.hindawi.com Advances in Journal of Hindawi Publishing Corporation ht p:/ www.hindawi.com Journal of bability and Statistics Hindawi Publishing Corporation ht p:/ www.hindawi.com Hindawi Publishing Corporation ht p:/ www.hindawi.com The Scientiifc World Journal Hindawi Publishing Corporation ht p:/ www.hindawi.com Combinatorics Hindawi Publishing Corporation ht p:/ www.hindawi.com Submit your manuscr ipts http://www.hindawi.com Differential Equatio Journal of Mathematics Hindawi Publishing Corporation ht p:/ www.hindawi.com En Hindawi Publishing Corporation ht p:/ www.hindawi.com Journal of Mathematics and Mathematical Sciences Journal of Discrete Mathematics ht p:/ w w.hindawi.com Journal of Function Spaces 2. KENNEDY , R.E. and COOPER , C.N. On the Natural Density of the Niven Numbers , College Math. Journal I_5 ( 1984 ), 309 - 312 . Volume 2014 Journal of Hindawi Publishing Corporation ht p:/ www .hindawi.com International Journal of (...truncated)


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Curtis N. Cooper, Robert E. Kennedy. A generalization of a theorem by Cheo and Yien concerning digital sums, International Journal of Mathematics and Mathematical Sciences, 9, DOI: 10.1155/S0161171286001011