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Scientia Magna Vol. 12 (2017), No. 1, 145-153 A survey on Smarandache notions in number theory II: pseudo-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 pseudo-Smarandache function. Keywords Smarandache notion, pseudo-Smarandache function, sequence, mean value. 2010 Mathematics Subject Classification 11A07, 11B50, 11L20, 11N25. §1. Definition and simple properties According to [11], the pseudo-Smarandache function Z(n) is defined by (cid:26) (cid:12)m(m+1)(cid:27) Z(n)=min m: n(cid:12) . (cid:12) 2 Some elementary properties can be found in [11] and [1]. R. Pinch [20]. For any given L > 0 there are infinitely many values of n such that Z(n+1) Z(n−1) >L, and there are infinitely many values of n such that >L. Z(n) Z(n) n For any integer k ≥2, the equation =k has infinitely many solutions n. Z(n) Z(2n) The ration is not bounded. Z(n) 1 Fix < β < 1 and integer t ≥ 5. The number of integers n with et−1 < n < et such that 2 Z(n)<nβ is at most 196t2eβt. ∞ (cid:88) 1 The series is convergent for any α>1. Z(n)α n=1 Some explicit expressions of Z(n) for some particular cases of n were given by Abdullah- Al-Kafi Majumdar. A. A. K. Majumdar [18]. If p≥5 is a prime, then   p−1, if 4|p−1, Z(2p) =  p, if 4|p+1,   p−1, if 3|p−1, Z(3p) =  p, if 3|p+1, 146 H.Liu No. 1   pp,−1, iiff 88||pp−+11,, Z(4p) =  33pp,−1, iiff 88||33pp++11,,   pp,−1, iiff 1122||pp−+11,, Z(6p) =  22pp,−1, iiff 44||33pp+−11,. A. A. K. Majumdar [18]. If p≥7 is a prime, then   pp,−1, iiff 1100||pp+−11,, Z(5p)=  22pp,−1, iiff 55||22pp+−11., If p≥11 is a prime, then   pp2p,−−1,1, iiifff 777|||pp2p−+−11,,1, Z(7p)=  323ppp,,−1, iiifff 577|||233ppp+−+111,,. If p≥13 is a prime, then  p−1, if 11|p−1,  p223ppp,,−−11,, iiiiffff 11111111||||p223ppp+−+−11,11,,, Z(11p)=  34455ppppp,,,−−11,, iiiiifffff 1111111111|||||34455ppppp+−+−+11111,,,,. Vol. 12 AsurveyonSmarandachenotionsinnumbertheoryII:pseudo-Smarandachefunction 147 A. A. K. Majumdar [18]. Let p and q be two primes with q >p≥5. Then Z(pq)=min{qy −1,px −1}, 0 0 where y = min{y : x,y ∈N,qy−px=1}, 0 x = min{x: x,y ∈N,px−qy =1}. 0 A. A. K. Majumdar [18]. If p ≥ 3 is a prime, then Z(2p2) = p2 −1. If p ≥ 5 is a prime, then Z(3p2)=p2−1. If p≥3 is a prime and k ≥3 is an integer, then   pk, if 4|p−1 and k is odd, Z(2pk) =  pk−1, otherwise,   pk, if 3|p+1 and k is odd, Z(3pk) =  pk−1, otherwise. S. Gou and J. Li [2]. The equation Z(n)=Z(n+1) has no positive integer solutions. For any given positive integer M, there exists a positive integer s such that |Z(s)−Z(s+1)|>M. Y. Zheng [29]. For any given positive integer M, there are infinitely many positive integers n such that |Z(n+1)−Z(n)|>M. M. Yang [27]. Suppose that n has primitive roots. Then Z(n) is a primitive root modulo n if and only if n=2,3,4. W. Lu, L. Gao, H. Hao and X. Wang [17]. Let p≥17 be a prime. Then we have Z(2p+1)≥10p, Z(2p−1)≥10p. L. Gao, H. Hao and W. Lu [?]. Let p≥17 be a prime, and let a,b be distinct positive integers. Then we have Z(ap+bp)≥10p. Y. Ji [10]. Let r be a positive integer. Suppose that r (cid:54)=1,2,3,5. Then 1(cid:16) (cid:112) (cid:17) Z(2r+1)≥ −1+ 2r+3·5+41 . 2 148 H.Liu No. 1 Assume that r (cid:54)=1,2,4,12. Then 1(cid:16) (cid:112) (cid:17) Z(2r−1)≥ −1+ 2r+3·3−23 . 2 §2. Mean values of the pseudo-Smarandache function Y. Lou [16]. For any real x>1 we have (cid:88) lnZ(n)=xlnx+O(x). n≤x W. Huang [9]. For any integer n>1 we have n (cid:88)lnZ(k) lnk (cid:18) (cid:19) (cid:18) (cid:19) 1 Z(n) 1 k=2 =1+O , =O . n lnn (cid:88)lnZ(k) lnn k≤n L. Cheng [4]. Let p(n) denote the smallest prime divisor of n, and let k be any fixed positive integer. For any real x>1 we have k (cid:18) (cid:19) (cid:88) p(n) = x +(cid:88) aix +O x , Z(n) lnx lnix lnk+1x n≤x i=2 where a (i=2,3,··· ,k) are computable constants. i X. Wang, L. Gao and W. Lu [23]. Define   0, if n=1, Ω(n)=  α1p1+α2p2+···+αrpr, if n=pα11pα22···pαrr. Let k ≥2 be any fixed positive integer. For any real x>1 we have (cid:88)Z(n)Ω(n)= ζ(3)x3 +(cid:88)k aix3 +O(cid:18) x3 (cid:19), 3lnx lnix lnk+1x n≤x i=2 where a (i=2,3,··· ,k) are computable constants. i H. Hao, L. Gao and W. Lu [8]. Let d(n) denote the divisor function, and let k ≥ 2 be any fixed positive integer. For any real x>1 we have (cid:88)Z(n)d(n)= π4 · x2 +(cid:88)k aix2 +O(cid:18) x2 (cid:19), 36 lnx lnix lnk+1x n≤x i=2 where a (i=2,3,··· ,k) are computable constants. i X. Wang, L. Gao and W. Lu [24]. Define (cid:40) m (cid:41) (cid:89) D(n)=min m: m∈N,n| d(i) . i=1 Vol. 12 AsurveyonSmarandachenotionsinnumbertheoryII:pseudo-Smarandachefunction 149 Let k ≥2 be any fixed positive integer. For any real x>1 we have (cid:88)Z(n)lnD(n)= ζ(3)ln2 · x3 +(cid:88)k aix3 +O(cid:18) x3 (cid:19), 3 lnx lnix lnk+1x n≤x i=2 where a (i=2,3,··· ,k) are computable constants. i §3. Thedualofthepseudo-Smarandachefunction,thenearpseudo-Smarandache function, and other generalizations According to [21], the dual of the pseudo-Smarandache function is defined by (cid:26) (cid:27) m(m+1) Z (n)=max m∈N: |n . ∗ 2 D. Liu and C. Yang [15]. Let A denote the set of simple numbers. For any real x≥1 we have (cid:88) x2 x2 (cid:18) x2 (cid:19) Z (n)=C +C +O , ∗ 1lnx 2ln2x ln3x n≤x n∈A where C ,C are computable constants. 1 2 X. Zhu and L. Gao [30]. We have ∞ ∞ (cid:88) Z∗(n) =ζ(α) (cid:88) 2m . nα mα(m+1)2α n=1 m=1 The near pseudo Smarandache function K(n) is defined as n (cid:88) K(n)= i+k(n), i=1 (cid:40) n (cid:41) (cid:88) wherek(n)=min k : k ∈N,n| i+k . SomerecurrenceformulassatisfiedbyK(n)were i=1 derived in [19]. H. Yang and R. Fu [26]. For any real x≥1 we have (cid:88)d(cid:18)K(n)− n(n+1)(cid:19) = 3xlogx+Ax+O(cid:16)x12 log2x(cid:17), 2 4 n≤x (cid:88)φ(cid:18)K(n)− n(n+1)(cid:19) = 93 x2+O(cid:16)x23+(cid:15)(cid:17), 2 28π2 n≤x where φ(n) denotes the Euler function, A is a computable constant, and (cid:15) > 0 is any real number. 1 Y. Zhang [28]. For any real number s> , the series 2 ∞ (cid:88) 1 Ks(n) n=1 150 H.Liu No. 1 is convergent, and ∞ ∞ (cid:88) 1 2 5 (cid:88) 1 11 22+2ln2 = ln2+ , = π2− . K(n) 3 6 K2(n) 108 27 n=1 n=1 Y. Li, R. Fu and X. Li [14]. We have (cid:88) x2lnlnx x2 2x2lnlnx (cid:18) x2 (cid:19) K(n) = +B + +O , 3lnx lnx 9ln2x ln2x n≤x n∈A (cid:18) (cid:19) (cid:88) 1 2 lnlnx = (lnlnx)2+Dlnlnx+E+O . K(n) 3 lnx n≤x n∈A L. Gao, R. Xie and Q. Zhao [5]. Define (cid:89) (cid:89) p (n)= d, q (n)= d. d d d|n d|n d<n For any real x>1 we have (cid:88) x5 x5 x5 (cid:18) x5 (cid:19) K(p (n)) = lnlnx+A + lnlnx+O , d 5lnx 1lnx 25ln2x ln2x n≤x n∈A (cid:88) x3 x3 x3 (cid:18) x3 (cid:19) K(q (n)) = lnlnx+A + lnlnx+O , d 3lnx 2lnx 9ln2x ln2x n≤x n∈A where A ,A are computable constants. 1 2 Other generalizations on the near pseudo-Smarandache function have been given. For example, define (cid:26) (cid:27) m(m+1)(m+2) Z (n)=min m: m∈N,n| . 3 6 The elementary properties were studied in [6] and [7]. Y. Wang [25]. Define U (n)=min(cid:8)k : 1t+2t+···+nt+k =m,n|m,k,t,m∈N(cid:9). t For any real number s>1, we have ∞ (cid:18) (cid:19) (cid:88) 1 1 = ζ(s) 2− , Us(n) 2s n=1 1 ∞ (cid:18) (cid:18) (cid:19)(cid:18) (cid:19)(cid:19) (cid:88) 1 1 1 1 1 = ζ(s) 1+ − +2 1− 1− , Us(n) 5s 6s 2s 3s n=1 2 (cid:88)∞ 1 (cid:32) (cid:18) 1 (cid:19)2(cid:33) = ζ(s) 1+ 1− . Us(n) 2s n=1 3 Vol. 12 AsurveyonSmarandachenotionsinnumbertheoryII:pseudo-Smarandachefunction 151 M. Tong [22]. Define   min{m: m∈N,n|m(m+1)}, if 2|n, Z (n)= 0  min{m: m∈N,n|m2}, if 2(cid:45)n. For any real x>1, we have (cid:88)Z0(2n−1)= 3ζπ(23)x2+O(cid:16)x32+(cid:15)(cid:17). n≤x X. Li [12]. Define (cid:26) (cid:27) a(a+1) C(n)=min a+b: a,b∈N,n| +b . 2 For any real x>1, we have √ (cid:88)C(n) = 2x32 +O(x), n≤x (cid:88) 1 √ = ln2· 2x+O(lnx), C(n) n≤x (cid:88)d(C(n)) = 1xlnx+x(cid:18)2γ+ 5ln2− 3(cid:19)+O(cid:16)x34(cid:17), 2 2 2 n≤x where γ is the Euler constant. Y. Li [13]. Define (cid:26) (cid:27) a(a+1) D(n)=max ab: a,b∈N,n= +b . 2 For any real x>1, we have √ (cid:88)D(n) = 4 6x25 +O(cid:0)x2(cid:1), 45 n≤x √ (cid:88) C(n) 9 3 = lnx+O(1). D(n) 4 n≤x References [1] Charles Ashbacher. Pluckings from the tree of Smarandache sequences and functions. American Research Press (ARP), Lupton, AZ, 1998. [2] Su Gou and Jianghua Li. On the pseudo-Smarandache function. 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