Given a positive integer n, let P(n) denote the largest prime factor of n and S(n) denote the smallest integer m such that n divides m

Finch, Steven R.

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THE A VERAGE VALUE OF THE SMARANDACHE FUNCTION Steven R. Finch MathSoft Inc. 101 Main Street Cambridge, MA, USA 02142 sfinch@mathsoft·com Given a positive integer n, let P(n) denote the largest prime factor of n and S(n) denote the smallest integer m such that n divides m! The function S(n) is known as the Smarandache function and has been intensively studied [1]. Its behavior is quite erratic [2] and thus all we can reasonably hope for is a statistical approximation of its growth, e.g., an average. It appears that the sample mean E(S) satisfies [3] 1·"i q N ) E(S(N» = -, . IS(n) = --, lv n=1 In(N) as l\' approaches infinity, but I don't know of a rigorous proof. A natural question is if some other sense of average might be more amenable to analysis. Erdos [4,5] pointed out that P(n) = S(n) for almost all n, meaning . I{n ~ N: pen) < S(n)}1 hm =0 S--+C() N that is, I{n ~ N: pen) < S(n)}1 = o(N) as N approaches infinity. Kastanas [5] proved this to be true, hence the following argument is valid. On one hand, , . (In(p(n))) . (In(S(n))) . 1. f In(S(n» /. = hm E ~ 11m E = hm - . ~ n--+C() In( n) n--+x \ In( n) :-; --+x j\i n=1 In( n) The above summation, on the other hand, breaks into 1\vo parts: . 1 ( ,,In(P(n)) ,,In(S(n)) ') hm-·l ~ + ~ j .\"--+00 N P(n)=S(n) In(n) P(n)<S(n) In(n) The second part vanishes: 95 while the first part is bounded from above: . 1 ( "" In(P(n)) '\ . 1...f In(P(n)) . (In(p(n))) hm-· ~ ~ hm-·~ =hmE =}. N ..... oo N lp(n)=s(n) In(n) ) N ..... oo N n=I In(n) n ..... oo In(n) We deduce that ( In(S(n))) lim EI = A = 0.6243299885 ... n ..... oo \ In(n) where A is the famous Golomb-Dickman constant [6-9]. Therefore A· n is the asymptotic average number of digits in the output of S at an n-digit input, that is, 62.43% of the original number of digits. As far as I know, this result about the Smarandache function has not been published before. A closely related unsolved problem concerns estimating the variance of S. References 1. C. Dumitrescu and V. Seleacu, The Smarandache Function, Erhus University Press, 1996. 2. N. 1. A. Sloane, On-Line Encyclopedia oj Integer Sequences, website URL http://www.research.att.coml-njas/sequences/.AT &T Research (sequences A002034 and A006530). 3. S. Tabirca and T. Tabrica, Some upper bounds ofSmarandache's function, Smarandache Notions Journal 8 (1997) 205-211. 4. P. Erdos, personal communication to T. Yau (1995), in Smarandache Notions Journal 8 (1997) 220-224. 5. 1. Kastanas, The smallest factorial that is a multiple of n, Amer. i\1ath. lvfonthly 101 (1994) 179. 6. L. A. Shepp and S. P. Lloyd, Ordered cycle lengths in a random permutation, Trans. Amer. Math. Soc. 121 (1966) 350-557. 7. D. E. Knuth and L. Trabb Pardo, Analysis of a simple factorization algorithm, Theoret. Compo Sci. 3 (1976) 321-348. 8. D. E. Knuth, The Art ojComputer Programming, vol. 2, Seminumerical Algorithms 2nd ed., Addison Wesley 1981 (pp. 367-368, 395 and 611). 9. S. R. Finch, Favorite Mathematical Constants, website URL http://W\Vw.mathsoft.comlasolve!constanticonstant.html. MathSoft Inc. 96 

THE A VERAGE VALUE OF THE SMARANDACHE FUNCTION

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