On the F. Smarandache function and its mean value

2007·0.24 MB·

by  Zhongtian Lv

Scientia Magna
Vol. 3 (2007), No. 2, 104-108
On the F.Smarandache function and its mean
value
Zhongtian Lv
Department of Basic Courses, Xi’an Medical College
Xi’an, Shaanxi, P.R.China
Received December 19, 2006
Abstract For any positive integer n, the famous F.Smarandache function S(n) is defined
asthesmallestpositiveintegermsuchthatn|m!. Thatis,S(n)=min{m: n|m!, n∈N}.
The main purpose of this paper is using the elementary methods to study a mean value
probleminvolvingtheF.Smarandachefunction,andgiveasharperasymptoticformulaforit.
Keywords F.Smarandache function, mean value, asymptotic formula.
§1. Introduction and result
For any positive integer n, the famous F.Smarandache function S(n) is defined as the
smallest positive integer m such that n | m!. That is, S(n) = min{m : n | m!, n ∈ N}. For
example, the first few values of S(n) are S(1) = 1, S(2) = 2, S(3) = 3, S(4) = 4, S(5) = 5,
S(6) = 3, S(7) = 7, S(8) = 4, S(9) = 6, S(10) = 5, ···. About the elementary properties of
S(n), some authors had studied it, and obtained some interesting results, see reference [2], [3]
and [4]. For example, Farris Mark and Mitchell Patrick [2] studied the elementary properties
of S(n), and gave an estimates for the upper and lower bound of S(pα). That is, they showed
that
(p−1)α+1≤S(pα)≤(p−1)[α+1+log α]+1.
p
Murthy [3] proved that if n be a prime, then SL(n) = S(n), where SL(n) defined as
the smallest positive integer k such that n | [1, 2, ··· , k], and [1, 2, ··· , k] denotes the
least common multiple of 1, 2, ···, k. Simultaneously, Murthy [3] also proposed the following
problem:
SL(n)=S(n), S(n)(cid:54)=n ? (1)
Le Maohua [4] completely solved this problem, and proved the following conclusion:
Every positive integer n satisfying (1) can be expressed as
n=12 or n=pα1pα2···pαrp,
1 2 r
where p , p , ···, p , p are distinct primes, and α , α , ···, α are positive integers satisfying
1 2 r 1 2 r
p>pαi, i=1, 2, ···, r.
i
Vol. 3 OntheF.Smarandachefunctionanditsmeanvalue 105
Dr. XuZhefeng[5]studiedthevaluedistributionproblemofS(n),andprovedthefollowing
conclusion:
Let P(n) denotes the largest prime factor of n, then for any real number x > 1, we have
the asymptotic formula
(cid:161) (cid:162) (cid:195) (cid:33)
(cid:88)(S(n)−P(n))2 = 2ζ 23 x32 +O x32 ,
3lnx ln2x
n≤x
where ζ(s) denotes the Riemann zeta-function.
On the other hand, Lu Yaming [6] studied the solutions of an equation involving the
F.Smarandache function S(n), and proved that for any positive integer k ≥2, the equation
S(m +m +···+m )=S(m )+S(m )+···+S(m )
1 2 k 1 2 k
has infinite groups positive integer solutions (m ,m ,··· ,m ).
1 2 k
Jozsef Sandor [7] proved for any positive integer k ≥2, there exist infinite groups positive
integer solutions (m , m , ··· , m ) satisfied the following inequality:
1 2 k
S(m +m +···+m )>S(m )+S(m )+···+S(m ).
1 2 k 1 2 k
Also, there exist infinite groups of positive integer solutions (m ,m ,··· ,m ) such that
1 2 k
S(m +m +···+m )<S(m )+S(m )+···+S(m ).
1 2 k 1 2 k
Themainpurposeofthispaperisusingtheelementaryandanalyticmethodstostudythe
mean value properties of [S(n)−S(S(n))]2, and give an interesting mean value formula for it.
That is, we shall prove the following conclusion:
Theorem. Let k be any fixed positive integer. Then for any real number x>2, we have
the asymptotic formula
(cid:181) (cid:182) (cid:195) (cid:33)
(cid:88)[S(n)−S(S(n))]2 = 2 ·ζ 3 ·x32 ·(cid:88)k ci +O x32 ,
3 2 lnix lnk+1x
n≤x i=1
where ζ(s) is the Riemann zeta-function, c (i = 1, 2,··· , k) are computable constants and
i
c =1.
1
§2. Proof of the Theorem
In this section, we shall prove our theorem directly. In fact for any positive integer n>1,
let n=pα1pα2···pαs be the factorization of n into prime powers, then from [3] we know that
1 2 s
S(n)=max{S(pα1), S(pα2), ··· , S(pαs)}≡S(pα). (2)
1 2 s
Now we consider the summation
(cid:88) (cid:88) (cid:88)
[S(n)−S(S(n))]2 = [S(n)−S(S(n))]2+ [S(n)−S(S(n))]2, (3)
n≤x n∈A n∈B
106 ZhongtianLv No. 2
where A and B denote the subsets of all positive integer in the interval [1, x]. A denotes the
set involving all integers n ∈ [1, x] such that S(n) = S(p2) for some prime p; B denotes the
set involving all integers n ∈ [1, x] such that S(n) = S(pα) with α = 1 or α ≥ 3. If n ∈ A,
then n = p2m with P(m) < 2p, where P(m) denotes the largest prime factor of m. So from
the definition of S(n) we have S(n)=S(mp2)=S(p2)=2p and S(S(n))=S(2p)=p if p>2.
From (2) and the definition of A we have
(cid:88)
[S(n)−S(S(n))]2
n∈A
(cid:88) (cid:163) (cid:164) (cid:88) (cid:163) (cid:164)
= S(p2)−S(S(p2)) 2+ S(p2)−S(S(p2)) 2
n≤x n≤x
√ √
p2(cid:107)n, n<p2 p2(cid:107)n, p2≤ n
(cid:88) (cid:163) (cid:164) (cid:88) (cid:163) (cid:164)
= S(p2)−S(S(p2)) 2+ S(p2)−S(S(p2)) 2
p2n≤x p2n≤x
n<p2, (p, n)=1 p2≤n, (p, n)=1
(cid:88) (cid:88)
= p2+ p2+O(1)
p2n≤x p2n≤x
n<p2, (p, n)=1 n≥p2, (p, n)=1
   
= (cid:88) (cid:88) p2+O (cid:88) (cid:88) p2+O (cid:88) (cid:88) p2
√
n≤ xn<p2≤nx m≤x41 p≤(x)13 p≤x41 p2≤n≤px2
(cid:195) (cid:33) m
= (cid:88) (cid:88) p2+O x54 , (4)
√ √ lnx
n≤ xp≤ x
n
where p2(cid:107)n denotes p2|n and p3†n.
By the Abel’s summation formula (See Theorem 4.2 of [8]) and the Prime Theorem (See
Theorem 3.2 of [9]):
(cid:181) (cid:182)
(cid:88)k a ·x x
π(x)= i +O ,
lnix lnk+1x
i=1
where a (i=1, 2,··· ,k) are computable constants and a =1.
i 1
We have
√
(cid:181)(cid:114) (cid:182) (cid:90)
(cid:88) p2 = x ·π x − nx 2y·π(y)dy
√ n n 3
p≤ x 2
n
(cid:195) (cid:33)
= 1 · x32 ·(cid:88)k b(cid:112)i +O x32 , (5)
3 n32 i=1 lni nx n32 ·lnk+1x
√
where we have used the estimate n≤ x, and all b are computable constants and b =1.
i 1
(cid:181) (cid:182)
(cid:88)∞ 1 3 (cid:88)∞ lnin
Note that =ζ , and is convergent for all i=1, 2, 3, ···, k. So from
n=1n32 2 n=1 n32
Vol. 3 OntheF.Smarandachefunctionanditsmeanvalue 107
(4) and (5) we have
(cid:88)
[S(n)−S(S(n))]2
n∈A
(cid:34) (cid:195) (cid:33)(cid:35) (cid:195) (cid:33)
= (cid:88) 1 · x32 ·(cid:88)k b(cid:112)i +O x32 +O x54
n≤√x 3 n32 i=1 lni nx n32 ·lnk+1x lnx
(cid:181) (cid:182) (cid:195) (cid:33)
= 2 ·ζ 3 ·x32 ·(cid:88)k ci +O x23 , (6)
3 2 lnix lnk+1x
i=1
where c (i=1,2, 3,··· ,k) are computable constants and c =1.
i 1
NowweestimatethesummationinsetB. Foranypositiveintegern∈B,ifS(n)=S(p)=
p, then [S(n)−S(S(n))]2 =[S(p)−S(S(p))]2 =0; If S(n)=S(pα) with α≥3, then
[S(n)−S(S(n))]2 =[S(pα)−S(S(pα))]2 ≤α2p2
and α≤lnx. So that we have
(cid:88) (cid:88)
[S(n)−S(S(n))]2 (cid:191) α2·p2 (cid:191)x·ln2x. (7)
n∈B npα≤x
α≥3
Combining (3), (6) and (7) we may immediately deduce the asymptotic formula
(cid:181) (cid:182) (cid:195) (cid:33)
(cid:88)[S(n)−S(S(n))]2 = 2 ·ζ 3 ·x32 ·(cid:88)k ci +O x32 ,
3 2 lnix lnk+1x
n≤x i=1
where c (i=1,2, 3,··· ,k) are computable constants and c =1.
i 1
This completes the proof of Theorem.
References
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1993.
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[3] A. Murthy, Some notions on least common multiples, Smarandache Notions Journal,
12(2001), 307-309.
[4] Le Maohua, An equation concerning the Smarandache LCM function, Smarandache
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108 ZhongtianLv No. 2
[6] Lu Yaming, On the solutions of an equation involving the Smarandache function, Sci-
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Magna, 2(2006), No. 3, 78-80.
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