ON AN ASYMPTOTIC FORMULA FOR THE NIVEN NUMBERS

A Niven number is a positive integer which is divisible by its digital sum. A discussion of the possibility of an asymptotic formula for N(x) is given. Here, N(x) denotes the nmber of Niven numbers less than x. A partial result will be presented. This result will be an asymptotic formula for Nk(x) which denotes the number of Niven numbers less than x with digital sum k.

ON AN ASYMPTOTIC FORMULA FOR THE NIVEN NUMBERS CURTIS N. COOPER and ROBERT E. KENNEDY Depar%zent of .ath,:aticsand Computer Science Central Missour State University Warrensburg, Missouri 64093 U.S.A.
(Received May 20,1985) ABSTRACT.A Niven number s a positive integer which is d.visible by its digital um.A d.sc.,ss.on of the possibiS.ity of an asymptotic formula for N(x) is given.
Here, N(x) denotes the nmber of Niven numbers less than x A partial result will be preented.This result wll be an asymptotic formula for Nk(X) which denotes the number of Niven numbers less than x with digital sum k },Y WOPDS AND PHRASES.Digital sums, asymptotic formula, Niven number.
In Kennedy et al [] the concept of a Niven number was introduced as any positive integer n which is divisible by its digital sum s(n) One of the first questions about the set, N of Niven numbers which was investigated was the status of L N(x) (.) x -x where N(x) denotes the number of Niven numbers less than x (In what follows, we will use the convention that if A is a set of integers, then A(x) will be the number of members of A less than x .)This limit, if it exists, is called the "natural density" of the set N Even %hough this was answered in Kenned and Cooper [2] (the natural densit? 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 ?
If such an f(x) exists, then the uual notation to indicate this is, 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 k may be written in the form

(.s)
In what.follows, we will develop an asymptotic formula for Nk(X ?.
A, ASY,:PTCIC FORMULA WHEN k 2asb c S,h a fo,u!a for Nk(X can easily found for k of the fo 2a503 c when c 0 or 2 This Is given in Theorem 2.1 with the help of the fo! owing emma. I..E:Y: I. Tet n be +ege.Then e, Nk(10 n) (2.!.) PRO'?, He, the --i,are brackets denote the atest integer function and %he pare,'nse ,enot, bnom.a7coeff$.e-ien%.;ote that an intec,r of the fom., f + ie(k) i=O where e i {0, I] c 0 '/ C + 02 + + Cf k and is a Niven number with digita] sum k But the sequence (2.) f (2.) can be rearranged exactly ways, and each of these will determine a Niven mmber with digital sum k T.h,:refore, we have that (2,1) hods.
Tt" 2. "* t:._ .wh.?", c , , or Then .;..() ,-,, (o)!: (2.) PROOF.Let , ":e the positive integer such that 10 n x < I0 n+l and so, not dependent on since each side of (2.10) and Nk(10n).#nk/k: (n,+l)k/k! ,Nk(i.On+l it readily follows that () ,,, (o )/: .o,=e n [ogx] o- .A LOW 0 0 ,() !t should " not. he that Lemma I can be used to de%eine a lower bound for N(x) In fact, the seah for such a lower dl to the meth that will give us an asptotic foula for Nk(X To deteine is lower bound, let k 2 m for some positive integer In what follows, for positive integers k and n we will denote the decimal represer.tation of x [0, iO n where [0, .I_0n) is the: set of no.-oegatve negers less than iO n as n-i xi10 i i=O Note that initial zeros will be allowed so that x will have n digits.For each j 0, _, 2, e(k)-we also define the finite sequences B(x,j) and T(x) x i x = y if and only if T(x) T(y) and B(x,j) is a reamngement of the trms of B(y,j) for each j [0 It is clear that = is an equivalence latlon on I0n) For x a memr of [0, 0n) let <x denote the ulvalence cass contalni x The fo].owlng ].ea wil us to help count the nr of Nien nm with I.E'W. .let x,.y 0n) Then x = v .:e:3.... that s(r, (v ., ) "s a reae-POOP.Z!:,oe x = y we hav tha% T(x) T(y) and j=o i=o j=o i=o (.8) m (.9) of digits, let for t O, I, 2, 9 Hee, the # symbol denotes the oadinality of the set.For example, if t then d 3 is the number of tems of the sequenoe equal to 3 Therefore, the number of finite sequences whleh cn be formed by rearranging the terms of (4.9) is given by the multinomial coefficient 0' dl' ""' d9 We will.use this fact to develop an asymptotic formula for I,EMNA 5. Let x N k [0, los) Then .#<x> of degree less than or equal to k where e(Ik N k(x) for any integer is a polynomial in f PROOF.Note that each y <x> may be found by rearranging the terms of B(x,j) for various j's Let dt(J) #{0 i f X(k) + j + le(k) t} (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_ / f .+' (4.1h.) jl= I o(j), d1(j), ..., d9(Jd where the sum is +.aken over the collection of equivalence classes induced by : N k EO, !0n) we have that Nk(10n is a polynomial in f of degree on not exceeding k by Lemma 5 Thus, all we need to do in ozder to show that Nk(!On has degree k is to constmct a Niven number, x, with digital sum k such that x> is a polynomlal in f of degree we have that f n/e() and therefore, Nk(lO n) cn k wher c cl/(e(k)) k-- Finally, us.ng a.n argument similiar to that in the proof of Theorem 2 we have the followng corollary COROLLARY 7. Let k, x be positive integers.Then Nk(x c(log x) k (.23) where c depends on k 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 found for caln integers k In fact, given a partic,]ar k it is indeed oossible to determine the exact form that c will be.This would involve an _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.