A survey on Smarandache notions in number theory I: Smarandache function

2017·0.25 MB·

by  Huaning Liu

Scientia Magna
Vol. 12 (2017), No. 1, 132-144
A survey on Smarandache notions in number
theory I: Smarandache function
Huaning Liu
School of Mathematics, Northwest University
Xi’an 710127, China
E-mail: [email protected]
Abstract In this paper we give a survey on recent results on Smarandache function.
Keywords Smarandache notion, Smarandache function, sequence, mean value.
2010 Mathematics Subject Classification 11A07, 11B50, 11L20, 11N25.
§1. Definition and simple properties
Foranypositiveintegern,thefamousSmarandachefunctionS(n)isdefinedasthesmallest
positive integer m such that n|m!. That is,
S(n)=min{m: n|m!, n∈N}. (1.1)
Many people studied the lower bound of S(n).
M. Le [16]. Let p>2 be a prime. Then S(cid:0)2p−1(2p−1)(cid:1)≥2p+1.
J. Su [35]. Let p≥5 be a prime. Then S(cid:0)2p−1(2p−1)(cid:1)≥6p+1.
J. Su and S. Shang [36]. Let p≥7 be a prime. Then S(2p+1)≥6p+1.
M. Liang [23]. Let p>7 be a prime. Then S(2p±1)≥8p+1.
T. Wen [40]. Let p≥17 be a prime. Then S(2p±1)≥10p+1.
C. Shi [33]. Let p≥17 be a prime. Then S(2p±1)≥14p+1.
X. Wang [38]. For any m∈N, let p≥9m2(logm+1)3 be a prime. Then S(2p−1)≥
2mp+1.
F. Li and C. Yang [20]. Let a and b be distinct positive integers, and let p ≥ 17 be a
prime. Then S(ap+bp)≥8p+1.
P. Shi and Z. Liu [34]. Let a and b be distinct positive integers, and let p ≥ 17 be a
prime. Then S(ap+bp)≥10p+1.
L. Gao, H. Hao and W. Lu [6]. Let a and b be positive integers with a > b, and let
p≥17 be a prime. Then S(ap−bp)≥8p+1.
J. Wang [37]. Let F = 22n +1 be the Fermat number. Then S(F ) ≥ 8·2n +1 for
n n
n≥3.
M. Zhu [55]. Let F = 22n +1 be the Fermat number. Then S(F ) ≥ 12·2n +1 for
n n
n≥3.
M. Liu and Y. Jin [26]. Let F = 22n +1 be the Fermat number. Then S(F ) ≥
n n
4(4n+9)·2n+1 for n≥4.
Vol. 12 AsurveyonSmarandachenotionsinnumbertheoryI:Smarandachefunction 133
M. Bencze [2]. For positive integer sequences m ,··· ,m , we have
1 n
(cid:32) n (cid:33) n
(cid:89) (cid:88)
S m ≤ S(m ).
k k
k=1 k=1
M. Le [17]. There are infinite many n∈N such that S(n)≤S(n−S(n)).
The distribution properties have also been studied.
W. Zhu [56]. Let m = pT1pT2···pTk, where p ,p ,··· ,p are distinct primes. For any
1 2 k 1 2 k
n∈N, we have
(cid:16) m (cid:17)
S(mn)=n· max {(p −1)T }+O lnn .
1≤i≤k i i lnm
M. Le [15]. For any distinct positive integers k and n, log S(cid:0)nk(cid:1) is never a positive
kn
integer.
F. Du [4]. 1. Assume that n = p p ···p , where p ,p ,··· ,p are distinct primes.
1 2 k 1 2 k
(cid:88) 1
Then can not be an integer.
S(d)
d|n
(cid:88) 1
2. Suppose that n=pT, where p>2 is a prime and T ≤p. Then can not be an
S(d)
d|n
integer.
3. Let n=pT1pT2···pTk−1·p , where p ,p ,··· ,p are distinct primes. If S(n)=p , then
1 2 k−1 k 1 2 k k
(cid:88) 1
can not be an integer.
S(d)
d|n
L. Huan [9]. 1. Assume that n = p p ···p , where p ,p ,··· ,p are distinct primes.
1 2 k 1 2 k
Then we have
(cid:89)S(d)=p ·p2···p2k−2p2k−1.
1 2 k−1 k
d|n
B. Liu and X. Pan [25]. For any positive integer n, the formula
S(2)S(4)···S(2n)
S(1)S(3)···S(2n−1)
is an integer if and only if n=1.
A. Zhang [49]. For integer n>1, we have
(cid:18) (cid:19)
1 1
|{m: 1≤m≤n,S(m) is a prime}|=1+O .
n lnn
W. Xiong [43]. Define
ES(n)=|{a:1≤a≤n,2|S(a)}|, OS(n)=|{a:1≤a≤n,2(cid:45)S(a)}|.
Then for integer n>1, we have
(cid:18) (cid:19)
ES(n) 1
=O .
OS(n) lnn
134 H.Liu No. 1
Q. Liao and W. Luo [24]. Let p be a prime and α be a positive integer.
1) For any positive integer r and α=pr, we have
S(pα)=pr+1−pr+p.
2) For any positive integer r, t∈[1,r] and α=pr−t, we have
S(pα)=pr+1−pr.
3) For any positive integer r, t∈[r+1,pr−pr−1] and α=pr−t.
(I) If
n−1
(cid:88)
α=pr−r− (−1)i−1(pki −k )+(−1)npkn
i
i=1
with
k <pki−1(p−1)−1, 1≤i≤n−1,
i
then we have
(cid:32) n (cid:33)
(cid:88)
S(pα)=(p−1) pr+ (−1)ipki +(−1)np.
i=1
(II) If
n−1
α=pr−r−(cid:88)(−1)i−1(pki −k )+(−1)n(cid:0)pkn −t(cid:1)
i
i=1
with t∈[1,k ] and
n
k <pki−1(p−1)−1, 1≤i≤n−1,
i
then
(cid:32) n (cid:33)
(cid:88)
S(pα)=(p−1) pr+ (−1)ipki .
i=1
Q. Liao and W. Luo [24]. Let φ(n) be the Euler function and let σ(n) be the sum of
the different positive factors for n.
1) For any positive integer k, there are no any prime p and positive integer m coprime with
p, such that φ(pm)=S(pk) and S(pk)≥S(mk).
2)Foranypositiveintegerk, iftherearesomeprimepandpositiveintegermcoprimewith
p, suchthatφ(p2m)=S(p2k)andS(p2k)≥S(mk), thenp=2k+1or2(cid:54)=p≤k. Furthermore,
(I) If 2k+1=p, then
(p,m)=(2k+1,1), (2k+1,2), (2,3).
(II) If 2≤p≤k, then k ≥3 and

 2≤φ(m)≤ 2k2+k−1, k ≡2(mod3),
3
 2≤φ(m)≤ 2k2+k, otherwise.
3
Vol. 12 AsurveyonSmarandachenotionsinnumbertheoryI:Smarandachefunction 135
3) For any positive integer k, if there are some prime p and positive integer m coprime
with p, such that φ(pαm) = S(pαk) and S(pαk) ≥ S(mk). Then αk+1 > pα−3(p2 −1) and
1≤φ(m)≤q, where
αk+1=qpα−3(p2−1)+r, 0≤r <pα−3(p2−1).
4) For any positive integer k, there exist some prime p and positive integer m coprime with
p, such that φ(p3m)=S(p3k) and S(p3k)≥S(mk), m=1,2.
Q. Liao and W. Luo [24]. 1) For any prime p, there is no any positive integer α such
σ(pα)
that is a positive integer.
S(pα)
2) Let p be an odd prime, α≥1 and n=2αp.
∞
(cid:88)(cid:104)p(cid:105) σ(n)
(I) If ≥α and is a positive integer, then 2α+1 ≡1(mod p).
2i S(n)
i=1
∞
(cid:88)(cid:104)p(cid:105) σ(n)
(II) If <α and is a positive integer, then
2i S(n)
i=1
σ(n) 2α+1−1 S(2α)
=m and p=m −1,
S(n) d d
where d=(cid:0)2α+1−1,S(2α)(cid:1) and 0<m≤d.
§2. Mean values of the Smarandache function
(cid:88)
C. Yang and D. Liu [45]. Define σ(n)= d. For any real x≥3 we have
d|n
(cid:88) π2 x2 (cid:18) x2 (cid:19)
σ(S(n))= · +O .
12 lnx ln2x
n≤x
Y. Wang [39]. For any real x≥2 we have the asymptotic formula
(cid:88) S(n) π2 x (cid:18) x (cid:19)
= · +O .
n 6 lnx ln2x
n≤x
W. Yao [48]. Let Λ(n) be the Mangoldt function. For any real x≥1 we have
(cid:88) x2 (cid:18)x2loglogx(cid:19)
Λ(n)S(n)= +O .
4 logx
n≤x
B. Shi [31]. Let k be any fixed positive integer. For any real x≥1 we have
(cid:88)Λ(n)S(n)=x2(cid:88)k ci +O(cid:18) x2 (cid:19),
logix logk+1x
n≤x i=0
where c (i=0,1,··· ,k) are constants, and c =1.
i 0
136 H.Liu No. 1
Z. Lv [28]. Let k be any fixed positive integer. For any real x>2 we have the asymptotic
formula
(cid:88)(S(n)−S(S(n)))2 = 2ζ(cid:18)3(cid:19)x23 (cid:88)k ci +O(cid:32) x32 (cid:33),
3 2 logix logk+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
J. Ge [7]. The Smarandache LCM function SL(n) is defined as the smallest positive
integer k such that n | [1,2,··· ,k], where [1,2,··· ,k] denotes the least common multiple of
1,2,··· ,k. Let k be any fixed positive integer. For any real x > 2 we have the asymptotic
formula
(cid:88)(SL(n)−S(n))2 = 2ζ(cid:18)3(cid:19)x23 (cid:88)k ci +O(cid:32) x32 (cid:33),
3 2 logix logk+1x
n≤x i=1
where ζ(s) is the Riemann zeta function, c (i=1,2,··· ,k) are computable constants.
i
X. Fan and C. Zhao [5]. Let d(n) be the divisor function. For any real x≥2 we have
(cid:88) π4 x2 (cid:18) x2 (cid:19)
S(n)d(n)= · +O .
36 lnx ln2x
n≤x
Z. Lv [29]. Let k ≥2 be any fixed positive integer. For any real x>1 we have
(cid:88)S(n)d(n)= π4 · x2 +(cid:88)k ci·x2 +O(cid:18) x2 (cid:19),
36 lnx lnix lnk+1x
n≤x i=2
where c (i=2,3,··· ,k) are computable constants.
i
(cid:88)
M. Zhu [54]. Define σ (n)= dα, α≥1. Let k ≥2 be any fixed positive integer. For
α
d|n
any real x>1 we have
(cid:88)S(n)σ (n)= ζ(α+2)ζ(2) · xα+2 +(cid:88)k ci·xα+2 +O(cid:18) xα+2 (cid:19),
α 2+α lnx lnix lnk+1x
n≤x i=2
where c (i=2,3,··· ,k) are computable constants.
i
H. Zhou [53]. Let k ≥1 be any fixed positive integer. For any complex s with Re s>1
we have
(cid:88)∞ Λ(nk) ζ(cid:48)(ks)
=−ζ(s) .
Ss(nk) ζ(ks)
n=1
Y. Guo [8]. Define a function F(n) as follows:

 0, if n=1,
F(n)=
 α1p1+α2p2+···+αrpr, if n>1 and n=pα11pα22···pαrr.
Let k ≥1 be any fixed positive integer. For any real x>1 we have
(cid:88)(F(n)−S(n))2 =(cid:88)k ci·x2 +O(cid:18) x2 (cid:19),
lni+1x lnk+2x
n≤x i=1
Vol. 12 AsurveyonSmarandachenotionsinnumbertheoryI:Smarandachefunction 137
π2
where c (i=1,2,··· ,k) are computable constants, and c = .
i 1 6
C. Shi [32]. For any positive integer k, the Smarandache kn-digital sequence a(k,n) is
defined as all positive integers which can be partitioned into two groups such that the second
part is k times bigger that the first. For 1≤k ≤9 and real x>1 we have
(cid:88) S(n) 3π2
= lnlnx+O(1).
a(k,n) 20k
n≤x
C. Yang, C. Li and D. Liu [44]. For any real x≥2 we have
(cid:88) ζ(3)x3 (cid:18) x3 (cid:19)
S2(n)= +O ,
3lnx ln2x
n≤x
(cid:88) S2(n) ζ(3)x2 (cid:18) x2 (cid:19)
= +O .
n 3lnx ln2x
n≤x
W. Huang [11]. Let k ≥1 be any fixed integer. For any real x≥2 we have
(cid:88) ζ(k+1) xk+1 (cid:18)xk+1(cid:19)
Sk(n)= · +O ,
k+1 lnx ln2x
n≤x
(cid:88) Sk(n) 2ζ(k+1) xk (cid:18) xk (cid:19)
= · +O .
n k+1 lnx ln2x
n≤x
C. Li, C. Yang and D. Liu [19]. Let P(n) denote the largest prime factor of n. For
any real x≥2 we have
(cid:32) (cid:33)
(cid:88) S(n) 6x23 x23
=xln2+ +O .
P(n) lnx ln2x
n≤x
M. Yang [46]. For any real x≥2 we have
(cid:18) (cid:19)
(cid:88) S(n) xlnlnx
=x+O ,
SL(n) lnx
n≤x
(cid:18) (cid:19)
(cid:88) P(n) xlnlnx
=x+O .
SL(n) lnx
n≤x
L. Li, J. Hao and R. Duan [22]. For any real x≥1 we have
(cid:88)
lnS(n)=xlnx+O(x).
n≤x
Z. Liu and P. Shi [27]. For any real x≥3 and β >1 we have
(cid:16) (cid:17)
(cid:88)(S(n)−P(n))β = 2ζ β+21 xβ+21 +O(cid:32)xβ+21(cid:33).
(β+1)lnx ln2x
n≤x
138 H.Liu No. 1
W. Huang [12]. For n =pα1pα2···pαk, we define (cid:36)(n)= p +p +···+p . For any
1 2 k 1 2 k
real x≥2 we have
(cid:88) x3 (cid:18) x3 (cid:19)
S(n)(cid:36)(n)=B +O ,
lnx ln2x
n≤x
where B is computable constant.
(cid:88)
G. Chen [3]. Define H(n) = S(r)S(s). Let k ≥ 1 be any fixed positive integer.
[r,s]=n
For any real x>1 we have
(cid:88)H(n)=(cid:88)k di·x3 +O(cid:18) x3 (cid:19),
lnix lnk+1x
n≤x i=1
1 ζ3(3)
where d (i=1,2,··· ,k) are computable constants, and d = · .
i 1 3 ζ(6)
Q. Yang [47]. For any real δ ≤1, the series
∞
(cid:88) 1
S(n)δ
n=1
diverges.
For any real (cid:15)>0, the series
∞
(cid:88) 1
S(n)(cid:15)S(n)
n=1
converges.
§3. Mean values of the Smarandache function over sequences
W. Zhang and Z. Xu [50]. Let a(n) denote the square complements of n. For any real
x≥3 we have the asymptotic formula
(cid:88) π2 x2 (cid:18) x2 (cid:19)
S(a(n))= · +O .
12 lnx ln2x
n≤x
H. Li and X. Zhao [21]. Let r (n) denote the integer part of k-th root of n. For any
k
real x≥3 we have
(cid:32) (cid:33)
(cid:88) π2 x1+k1 x1+k1
S(r (n))= · +O .
k 6(k+1) lnx ln2x
n≤x
J. Ma [30]. Define L(n)=[1,2,··· ,n]. For any real x≥1 we have
(cid:88)S(L(n))= 1x2+O(cid:16)x2138+(cid:15)(cid:17).
2
n≤x
Vol. 12 AsurveyonSmarandachenotionsinnumbertheoryI:Smarandachefunction 139
(cid:26) (cid:27)
k(k+1)
Q. Wu [41]. Define Z(n) = min k :n≤ . Let k ≥ 2 be any fixed positive
2
integer. For any real x>1 we have
(cid:88)S(Z(n))= π2 · (2x)32 +(cid:88)k ci(2x)23 +O(cid:32) x32 (cid:33),
18 ln2x lni2x lnk+1x
n≤x i=2
where c (i=2,3,··· ,k) are computable constants.
i
H. Zhao [51]. Let a (n) denote the k-th power complements of n. For any real x ≥ 3
k
we have
(cid:88)(S(a (n))−(k−1)P(n))2 = 2ζ(cid:0)23(cid:1) · x32 +O(cid:32) x32 (cid:33).
k 3 lnx ln2x
n≤x
W. Huang [10]. Define u(n)=min{k :n≤k(2k−1)}. Let k ≥2 be any fixed positive
integer. For any real x>1 we have
(cid:88)S(u(n))= π2 · (2√x)32 +(cid:88)k ci(2√x)23 +O(cid:32) x32 (cid:33),
144 ln 2x lni 2x lnk+1x
n≤x i=2
where c (i=2,3,··· ,k) are computable constants.
i
Q. Zhao and L. Gao [52]. Define W(n) = min{k :n≤k(3k+1)}. Let k ≥ 2 be any
fixed positive integer. For any real x>1 we have
(cid:88)S(W(n))= π2 · (3√x)32 +(cid:88)k bi(3√x)23 +O(cid:32) x32 (cid:33),
486 ln 3x lni 3x lnk+1x
n≤x i=2
where b (i=2,3,··· ,k) are computable constants.
i
W. Huang and J. Zhao [14]. Define
(cid:26) (cid:27)
1 1
u (n) = min m+ m(m−1)(r−2):n≤m+ m(m−1)(r−2),r ∈N,r ≥3 ,
r 2 2
(cid:26) (cid:27)
1 1
v (n) = max m+ m(m−1)(r−2):n≥m+ m(m−1)(r−2),r ∈N,r ≥3 .
r 2 2
Let k ≥2 be any fixed positive integer. For any real x>1 we have
(cid:88)S(u (n)) = π2 · (2(r−2)x)32 +(cid:88)k ci(2(r−2)x)23 +O(cid:32) x32 (cid:33),
r 18(r−2)3 ln(cid:112)2(r−2)x lni(cid:112)2(r−2)x lnk+1x
n≤x i=2
(cid:88)S(v (n)) = π2 · (2(r−2)x)32 +(cid:88)k ci(2(r−2)x)23 +O(cid:32) x32 (cid:33),
r 18(r−2)3 ln(cid:112)2(r−2)x lni(cid:112)2(r−2)x lnk+1x
n≤x i=2
where c (i=2,3,··· ,k) are computable constants.
i
W. Huang [13]. Define a(n)=n−u (n) and b(n)=v (n)−n. Let k ≥1 be any fixed
r r
positive integer. For any real x>1 we have
√ (cid:32) (cid:33)
(cid:88) 842π2 x74 x74
S(n)a(n) = · +O ,
63(r−2)54 ln2x ln22x
n≤x
140 H.Liu No. 1
√ (cid:32) (cid:33)
(cid:88) 842π2 x74 x74
S(n)b(n) = · +O .
63(r−2)54 ln2x ln22x
n≤x
(cid:89)
R. Xie, L. Gao and Q. Zhao [42]. Define q (n)= . Let k ≥1 be any fixed positive
d
d|n
d<n
integer. For any real x>1 we have
(cid:88)(cid:18) (cid:18)1 (cid:19) (cid:19)2 (cid:88)k x23 (cid:32) x32 (cid:33)
S(q (n))− d(n)−1 P(n) = c +O ,
d 2 ilnix lnk+1x
n≤x i=1
where c (i=1,2,··· ,k) are computable constants, and
i
3 ζ4(cid:0)3(cid:1) (cid:18)3(cid:19) 2 (cid:18)3(cid:19)
c = · 2 −2ζ2 + ζ .
1 2 ζ(3) 2 3 2
B. Li, J. Guo and H. Dong [18]. Define

 1, if n=1,
U(n)=
 max{α1p1,α2p2,··· ,αrpr}, if n=pα11pα22···pαrr.
1≤i≤r
Let k ≥2 be any fixed positive integer. For any real x≥3 we have
(cid:88)(S(a (n))−(k−1)U(n))2 = 2ζ(cid:18)3(cid:19)k2· x32 +O(cid:32) x161 (cid:33).
k 3 2 lnx ln2x
n≤x
J. Bai and W. Huang [1]. Let A denote the set of the simple numbers. Let k ≥ 2 be
any fixed positive integer. For any real x≥2 we have
(cid:88)Sk(n)= Bxk+1 +(cid:88)k Cixk+1 +O(cid:18) xk+1 (cid:19),
(k+1)lnx lnix lnk+1x
n≤x i=2
n∈A
√ √
(cid:18) (cid:19)
(cid:88) 1 E xlnlnx x
=Dlnlnx+ +O ,
S(n) lnx lnx
n≤x
n∈A
where B,D,E, C (i=2,3,··· ,k) are computable constants.
i
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