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: hnliu@nwu.edu.cn 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 For any positive integer n, the famous 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} .
(1.1)
Many people studied the lower bound of S(n). M. Le [16]. Let p > 2 be a prime. Then S 2p−1 (2p − 1) ≥ 2p + 1. J. Su [35]. Let p ≥ 5 be a prime. Then S 2p−1 (2p − 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 (log m + 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. n J. Wang [37]. Let Fn = 22 + 1 be the Fermat number. Then S(Fn ) ≥ 8 · 2n + 1 for n ≥ 3. n M. Zhu [55]. Let Fn = 22 + 1 be the Fermat number. Then S(Fn ) ≥ 12 · 2n + 1 for n ≥ 3. n M. Liu and Y. Jin [26]. Let Fn = 22 + 1 be the Fermat number. Then S(Fn ) ≥ 4(4n + 9) · 2n + 1 for n ≥ 4.
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M. Bencze [2]. For positive integer sequences m1 , · · · , mn , we have ! n n Y X S mk ≤ S (mk ) . 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 = pT1 1 pT2 2 · · · pkTk , where p1 , p2 , · · · , pk are distinct primes. For any n ∈ N, we have m S (mn ) = n · max {(pi − 1)Ti } + O ln n . 1≤i≤k ln m M. Le [15]. For any distinct positive integers k and n, logkn S nk is never a positive integer. F. Du [4]. 1. Assume that n = p1 p2 · · · pk , where p1 , p2 , · · · , pk are distinct primes. X 1 can not be an integer. Then S(d) d|n X 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. Tk−1 3. Let n = pT1 1 pT2 2 · · · pk−1 · pk , where p1 , p2 , · · · , pk are distinct primes. If S(n) = pk , then X 1 can not be an integer. S(d) d|n
L. Huan [9]. Then we have
1. Assume that n = p1 p2 · · · pk , where p1 , p2 , · · · , pk are distinct primes. Y
k−2
k−1
S(d) = p1 · p22 · · · p2k−1 p2k
.
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 1 |{m : 1 ≤ m ≤ n, S(m) is a prime}| = 1 + O n
1 ln n
.
W. Xiong [43]. Define ES(n) = |{a : 1 ≤ a ≤ n, 2 | S(a)}| ,
OS(n) = |{a : 1 ≤ a ≤ n, 2 - S(a)}| .
Then for integer n > 1, we have ES(n) =O OS(n)
1 ln n
.
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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 X α = pr − r − (−1)i−1 (pki − ki ) + (−1)n pkn i=1
with ki < pki −1 (p − 1) − 1, then we have S(pα ) = (p − 1) pr +
1 ≤ i ≤ n − 1,
n X
! (−1)i pki
+ (−1)n p.
i=1
(II) If α = pr − r −
n−1 X
(−1)i−1 (pki − ki ) + (−1)n pkn − t
i=1
with t ∈ [1, kn ] and ki < pki −1 (p − 1) − 1, then
1 ≤ i ≤ n − 1,
n X S(p ) = (p − 1) p + (−1)i pki α
r
! .
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) For any positive integer k, if there are some prime p and positive integer m coprime with p, such that φ(p2 m) = S(p2k ) and S(p2k ) ≥ S(mk ), then p = 2k +1 or 2 6= p ≤ k. Furthermore, (I) If 2k + 1 = p, then (p, m) = (2k + 1, 1), (II) If 2 ≤ p ≤ k, then k ≥ 3 and 2 ≤ φ(m) ≤ 2 ≤ φ(m) ≤
(2k + 1, 2),
2k2 +k−1 , 3 2k2 +k , 3
(2, 3).
k ≡ 2(mod3), otherwise.
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3) For any positive integer k, if there are some prime p and positive integer m coprime with p, such that φ(pα m) = 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(p3 m) = 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 = 2α p. ∞ h X σ(n) pi ≥ α and is a positive integer, then 2α+1 ≡ 1(mod p). (I) If i 2 S(n) i=1 ∞ h X σ(n) pi < α and (II) If is a positive integer, then i 2 S(n) i=1 σ(n) 2α+1 − 1 =m S(n) d
and
p=m
S(2α ) − 1, d
where d = 2α+1 − 1, S(2α ) and 0 < m ≤ d. §2. Mean values of the Smarandache function X C. Yang and D. Liu [45]. Define σ(n) = d. For any real x ≥ 3 we have d|n
X n≤x
π 2 x2 · +O σ (S(n)) = 12 ln x
x2 ln2 x
.
Y. Wang [39]. For any real x ≥ 2 we have the asymptotic formula X S(n) π2 x x = · +O . n 6 ln x ln2 x
n≤x
W. Yao [48]. Let Λ(n) be the Mangoldt function. For any real x ≥ 1 we have X n≤x
x2 Λ(n)S(n) = +O 4
x2 log log x log x
.
B. Shi [31]. Let k be any fixed positive integer. For any real x ≥ 1 we have X n≤x
Λ(n)S(n) = x
2
k X i=0
ci +O logi x
where ci (i = 0, 1, · · · , k) are constants, and c0 = 1.
x2 logk+1 x
,
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Z. Lv [28]. Let k be any fixed positive integer. For any real x > 2 we have the asymptotic formula ! 3 k X X 3 2 3 ci x2 2 (S(n) − S (S(n))) = ζ , x2 +O 3 2 logi x logk+1 x i=1 n≤x
where ζ(s) is the Riemann zeta function, ci (i = 1, 2, · · · , k) are computable constants, and c1 = 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 ! 3 k X X 3 2 3 x2 ci 2 (SL(n) − S(n)) = ζ x2 +O , 3 2 logi x logk+1 x i=1 n≤x where ζ(s) is the Riemann zeta function, ci (i = 1, 2, · · · , k) are computable constants. X. Fan and C. Zhao [5]. Let d(n) be the divisor function. For any real x ≥ 2 we have 2 X π 4 x2 x S(n)d(n) = · +O . 36 ln x ln2 x n≤x Z. Lv [29]. Let k ≥ 2 be any fixed positive integer. For any real x > 1 we have k
X
S(n)d(n) =
n≤x
X ci · x2 π 4 x2 · + +O 36 ln x i=2 lni x
x2
,
lnk+1 x
where ci (i = 2, 3, · · · , k) are computable X constants. M. Zhu [54]. Define σα (n) = dα , α ≥ 1. Let k ≥ 2 be any fixed positive integer. For d|n
any real x > 1 we have k
X
S(n)σα (n) =
n≤x
ζ(α + 2)ζ(2) xα+2 X ci · xα+2 · + +O 2+α ln x lni x i=2
xα+2 lnk+1 x
,
where ci (i = 2, 3, · · · , k) are computable constants. H. Zhou [53]. Let k ≥ 1 be any fixed positive integer. For any complex s with Re s > 1 we have ∞ X Λ(nk ) ζ 0 (ks) = −ζ(s) . S s (nk ) ζ(ks) n=1 Y. Guo [8]. Define a function F (n) as follows: 0, if n = 1, F (n) = α1 p1 + α2 p2 + · · · + αr pr , if n > 1 and n = pα1 pα2 · · · pαr . r 1 2 Let k ≥ 1 be any fixed positive integer. For any real x > 1 we have X n≤x
2
(F (n) − S(n)) =
k X ci · x2 x2 + O , lni+1 x lnk+2 x i=1
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π2 . where ci (i = 1, 2, · · · , k) are computable constants, and c1 = 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 X S(n) 3π 2 = ln ln x + O (1) . a(k, n) 20k
n≤x
C. Yang, C. Li and D. Liu [44]. For any real x ≥ 2 we have 3 X ζ(3)x3 x 2 , S (n) = +O 3 ln x ln2 x n≤x 2 X S 2 (n) ζ(3)x2 x = +O . n 3 ln x ln2 x n≤x W. Huang [11]. Let k ≥ 1 be any fixed integer. For any real x ≥ 2 we have k+1 X x ζ(k + 1) xk+1 k · +O S (n) = , k+1 ln x ln2 x n≤x k X S k (n) 2ζ(k + 1) xk x = · +O . n k+1 ln x ln2 x n≤x C. Li, C. Yang and D. Liu [19]. any real x ≥ 2 we have
Let P (n) denote the largest prime factor of n. For
2 X S(n) 6x 3 = x ln 2 + +O P (n) ln x
n≤x
2
x3 ln2 x
! .
M. Yang [46]. For any real x ≥ 2 we have X S(n) x ln ln x =x+O , SL(n) ln x n≤x X P (n) x ln ln x =x+O . SL(n) ln x n≤x
L. Li, J. Hao and R. Duan [22]. For any real x ≥ 1 we have X ln S(n) = x ln x + O (x) . n≤x
Z. Liu and P. Shi [27]. For any real x ≥ 3 and β > 1 we have β+1 ! β+1 x 2 2ζ β+1 X 2 x 2 β +O . (S(n) − P (n)) = (β + 1) ln x ln2 x n≤x
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ιk 1 ι2 W. Huang [12]. For n = pι 1 p2 ¡ ¡ ¡ pk , we define $(n) = p1 + p2 + ¡ ¡ ¡ + pk . For any real x ≼ 2 we have 3 X x3 x , S(n)$(n) = B +O ln x ln2 x n≤x
where B is computable constant. X 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 X
H(n) =
n≤x
k X di ¡ x3 i=1
lni x
+O
x3
lnk+1 x
where di (i = 1, 2, ¡ ¡ ¡ , k) are computable constants, and d1 =
,
1 Μ 3 (3) ¡ . 3 Μ(6)
Q. Yang [47]. For any real δ ≤ 1, the series ∞ X
1 δ S(n) n=1 diverges. For any real > 0, the series ∞ X
1 S(n) 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 2 X π 2 x2 x S(a(n)) = ¡ +O . 12 ln x ln2 x n≤x H. Li and X. Zhao [21]. Let rk (n) denote the integer part of k-th root of n. For any real x ≼ 3 we have ! 1 1 X π2 x1+ k x1+ k ¡ +O S(rk (n)) = . 6(k + 1) ln x ln2 x n≤x J. Ma [30]. Define L(n) = [1, 2, ¡ ¡ ¡ , n]. For any real x ≼ 1 we have X n≤x
S (L(n)) =
23 1 2 x + O x 18 + . 2
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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 ! 3 3 3 k X X x2 π 2 (2x) 2 ci (2x) 2 , +O · + S (Z(n)) = 18 ln 2x lni 2x lnk+1 x i=2 n≤x where ci (i = 2, 3, · · · , k) are computable constants. H. Zhao [51]. Let ak (n) denote the k-th power complements of n. For any real x ≥ 3 we have ! 3 3 X 2ζ 23 x2 x2 2 (S(ak (n)) − (k − 1)P (n)) = . · +O 3 ln x ln2 x 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 ! 3 3 3 k X X x2 π 2 (2x) 2 ci (2x) 2 √ +O , S (u(n)) = · √ + 144 ln 2x i=2 lni 2x lnk+1 x n≤x where ci (i = 2, 3, · · · , k) are computable constants. 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 ! 3 3 3 k X X bi (3x) 2 π 2 (3x) 2 x2 √ +O · √ + , S (W (n)) = 486 ln 3x i=2 lni 3x lnk+1 x n≤x where bi (i = 2, 3, · · · , k) are computable constants. W. Huang and J. Zhao [14]. Define 1 1 ur (n) = min m + m(m − 1)(r − 2) : n ≤ m + m(m − 1)(r − 2), r ∈ N, r ≥ 3 , 2 2 1 1 vr (n) = max m + m(m − 1)(r − 2) : n ≥ m + m(m − 1)(r − 2), r ∈ N, r ≥ 3 . 2 2 Let k ≥ 2 be any fixed positive integer. For any real x > 1 we have 3
X
S(ur (n))
=
X ci (2(r − 2)x) 2 π2 (2(r − 2)x) 2 p + +O · p 3 18(r − 2) ln 2(r − 2)x i=2 lni 2(r − 2)x
3
S(vr (n))
3
=
n≤x
X
k
X ci (2(r − 2)x) 2 π2 (2(r − 2)x) 2 p · p + +O 3 18(r − 2) ln 2(r − 2)x i=2 lni 2(r − 2)x
n≤x
k
3
3
x2
! ,
lnk+1 x 3
x2 lnk+1 x
! ,
where ci (i = 2, 3, · · · , k) are computable constants. W. Huang [13]. Define a(n) = n − ur (n) and b(n) = vr (n) − n. Let k ≥ 1 be any fixed positive integer. For any real x > 1 we have ! √ 7 7 X 8 4 2π 2 x4 x4 S(n)a(n) = +O , 5 · ln2 2x 63(r − 2) 4 ln 2x n≤x
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X
S(n)b(n)
=
n≤x
No. 1
√ 8 4 2π 2
7
7
x4 +O 5 · 63(r − 2) 4 ln 2x
R. Xie, L. Gao and Q. Zhao [42]. Define qd (n) =
Y
x4 ln2 2x
! .
. Let k ≥ 1 be any fixed positive
d|n d<n
integer. For any real x > 1 we have X
S (qd (n)) −
n≤x
2 X 3 k 1 x2 d(n) − 1 P (n) = ci i + O 2 ln x i=1
!
3
x2
,
lnk+1 x
where ci (i = 1, 2, · · · , k) are computable constants, and 2 3 3 3 ζ 4 32 − 2ζ 2 + ζ . c1 = · 2 ζ(3) 2 3 2 B. Li, J. Guo and H. Dong [18]. Define 1, if n = 1, U (n) = αr 1 α2 max {α1 p1 , α2 p2 , · · · , αr pr }, if n = pα 1 p2 · · · pr . 1≤i≤r
Let k ≥ 2 be any fixed positive integer. For any real x ≥ 3 we have X n≤x
2 (S (ak (n)) − (k − 1) U (n)) = ζ 3 2
3 3 x2 +O k2 · 2 ln x
11
x6 ln2 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 k
X n≤x n∈A
X n≤x n∈A
S k (n) =
X Ci xk+1 Bxk+1 + +O (k + 1) ln x i=2 lni x
xk+1 lnk+1 x
,
√ √ 1 E x ln ln x x = D ln ln x + +O , S(n) ln x ln x
where B, D, E, Ci (i = 2, 3, · · · , k) are computable constants.
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