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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 Xiâ€™an 710127, China E-mail: hnliu@nwu.edu.cn 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.

Â§1. Definition and simple properties According to [11], the pseudo-Smarandache function Z(n) is defined by

m(m + 1)

. Z(n) = min m : n

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 âˆ’ 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 â‰Ľ 2, the equation = k has infinitely many solutions n. Z(n) Z(2n) is not bounded. The ration Z(n) 1 Fix < Î˛ < 1 and integer t â‰Ľ 5. The number of integers n with etâˆ’1 < n < et such that 2 Z(n) < nÎ˛ is at most 196t2 eÎ˛t . âˆž X 1 is convergent for any Îą > 1. The series Z(n)Îą n=1 Some explicit expressions of Z(n) for some particular cases of n were given by AbdullahAl-Kafi Majumdar. A. A. K. Majumdar [18]. If p â‰Ľ 5 is a prime, then ďŁą ďŁ˛ p âˆ’ 1, Z(2p) = ďŁł p, ďŁą ďŁ˛ p âˆ’ 1, Z(3p) = ďŁł p,

if 4 | p âˆ’ 1, if 4 | p + 1, if 3 | p âˆ’ 1, if 3 | p + 1,

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  p − 1,      p, Z(4p) =   3p − 1,     3p,   p − 1,      p, Z(6p) =   2p − 1,     2p,

A. A. K. Majumdar [18]. If p ≥ 7 is a   p − 1,      p, Z(5p) =   2p − 1,     2p,

No. 1

if 8 | p − 1, if 8 | p + 1, if 8 | 3p + 1, if 8 | 3p + 1, if 12 | p − 1, if 12 | p + 1, if 4 | 3p + 1, if 4 | 3p − 1.

prime, then if 10 | p − 1, if 10 | p + 1, if 5 | 2p − 1, if 5 | 2p + 1.

If p ≥ 11 is a prime, then   p − 1,       p,      2p − 1, Z(7p) =   2p,      3p − 1,      3p,

if 7 | p − 1, if 7 | p + 1, if 7 | 2p − 1, if 5 | 2p + 1, if 7 | 3p − 1, if 7 | 3p + 1.

If p ≥ 13 is a prime, then

Z(11p) =

                       

p − 1,

if 11 | p − 1,

if 11 | p + 1,

                      

2p − 1, if 11 | 2p − 1, 2p,

if 11 | 2p + 1,

3p − 1,

if 11 | 3p − 1,

3p,

if 11 | 3p + 1,

4p − 1, if 11 | 4p − 1, 4p,

if 11 | 4p + 1,

5p − 1, if 11 | 5p − 1, 5p,

if 11 | 5p + 1.

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A. A. K. Majumdar [18]. Let p and q be two primes with q > p ≥ 5. Then Z(pq) = min {qy0 − 1, px0 − 1} , where y0

min {y : x, y ∈ N, qy − px = 1} ,

min {x : x, y ∈ N, px − qy = 1} .

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]. integers n such that

For any given positive integer M , there are infinitely many positive |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 6= 1, 2, 3, 5. Then Z (2r + 1) ≥

p 1 −1 + 2r+3 · 5 + 41 . 2

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No. 1

Assume that r 6= 1, 2, 4, 12. Then Z (2r − 1) ≥

p 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 X ln Z(n) = x ln x + O(x). n≤x

W. Huang [9]. For any integer n > 1 we have n X ln Z(k) k=2

ln k

=1+O

1 ln n

Z(n) X =O ln Z(k)

1 ln n

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 X p(n) X ai x x x = + +O , Z(n) ln x i=2 lni x lnk+1 x

n≤x

where ai (i = 2, 3, · · · , k) are computable constants. X. Wang, L. Gao and W. Lu [23]. Define   0, if n = 1, Ω(n) =  α1 p1 + α2 p2 + · · · + αr pr , if n = pα1 pα2 · · · pαr . r 1 2 Let k ≥ 2 be any fixed positive integer. For any real x > 1 we have k

X n≤x

Z(n)Ω(n) =

ζ(3)x3 X ai x3 + +O 3 ln x lni x i=2

lnk+1 x

where ai (i = 2, 3, · · · , k) are computable constants. 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 k

X n≤x

X ai x2 π 4 x2 Z(n)d(n) = · + +O 36 ln x i=2 lni x

where ai (i = 2, 3, · · · , k) are computable constants. X. Wang, L. Gao and W. Lu [24]. Define ( D(n) = min m : m ∈ N, n |

m Y i=1

lnk+1 x

) d(i) .

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Let k â‰Ľ 2 be any fixed positive integer. For any real x > 1 we have k

Z(n) ln D(n) =

nâ‰¤x

X ai x3 Îś(3) ln 2 x3 Âˇ + +O 3 ln x i=2 lni x

x3 lnk+1 x

where ai (i = 2, 3, Âˇ Âˇ Âˇ , k) are computable constants. Â§3. The dual of the pseudo-Smarandache function, the near pseudo-Smarandache function, and other generalizations According to [21], the dual of the pseudo-Smarandache function is defined by m(m + 1) Zâˆ— (n) = max m âˆˆ N : |n . 2 D. Liu and C. Yang [15]. Let A denote the set of simple numbers. For any real x â‰Ľ 1 we have 2 X x x2 x2 Zâˆ— (n) = C1 + C2 2 + O , ln x ln x ln3 x nâ‰¤x nâˆˆA

where C1 , C2 are computable constants. X. Zhu and L. Gao [30]. We have âˆž âˆž X X Zâˆ— (n) 2m = Îś(Îą) . Îą Îą (m + 1)2Îą n m n=1 m=1

The near pseudo Smarandache function K(n) is defined as K(n) =

n X

i + k(n),

i=1

( where k(n) = min k : k âˆˆ N, n |

n X

) i + k . Some recurrence formulas satisfied by K(n) were

i=1

derived in [19]. H. Yang and R. Fu [26]. For any real x â‰Ľ 1 we have 1 X 3 n(n + 1) = x log x + Ax + O x 2 log2 x , d K(n) âˆ’ 2 4 nâ‰¤x 3 X n(n + 1) 93 2 Ď† K(n) âˆ’ = x + O x 2 + , 2 2 28Ď€ nâ‰¤x

where Ď†(n) denotes the Euler function, A is a computable constant, and > 0 is any real number. 1 Y. Zhang [28]. For any real number s > , the series 2 âˆž X

1 s (n) K n=1

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No. 1

is convergent, and âˆž X

âˆž X

1 2 5 = ln 2 + , K(n) 3 6 n=1

n=1

1 K 2 (n)

11 2 22 + 2 ln 2 Ď€ âˆ’ . 108 27

Y. Li, R. Fu and X. Li [14]. We have X

K(n)

x2 2x2 ln ln x x2 ln ln x +O +B + 3 ln x ln x 9 ln2 x

2 (ln ln x)2 + D ln ln x + E + O 3

nâ‰¤x nâˆˆA

X nâ‰¤x nâˆˆA

1 K(n)

x2 ln2 x

ln ln x ln x

L. Gao, R. Xie and Q. Zhao [5]. Define pd (n) =

qd (n) =

d|n

d|n d<n

For any real x > 1 we have X

K (pd (n))

x5 x5 x5 ln ln x + A1 + ln ln x + O 5 ln x ln x 25 ln2 x

x3 x3 x3 ln ln x + A2 + ln ln x + O 3 ln x ln x 9 ln2 x

nâ‰¤x nâˆˆA

K (qd (n))

nâ‰¤x nâˆˆA

x5 ln2 x

x3 ln2 x

where A1 , A2 are computable constants. Other generalizations on the near pseudo-Smarandache function have been given. For example, define m(m + 1)(m + 2) Z3 (n) = min m : m âˆˆ N, n | . 6 The elementary properties were studied in [6] and [7]. Y. Wang [25]. Define Ut (n) = min k : 1t + 2t + Âˇ Âˇ Âˇ + nt + k = m, n | m, k, t, m âˆˆ N . For any real number s > 1, we have âˆž X

1 s (n) U n=1 1 âˆž X

U s (n) n=1 2 âˆž X 1 s (n) U n=1 3

1 = Îś(s) 2 âˆ’ s , 2 1 1 1 1 = Îś(s) 1 + s âˆ’ s + 2 1 âˆ’ s 1âˆ’ s , 5 6 2 3 2 ! 1 = Îś(s) 1 + 1 âˆ’ s . 2

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M. Tong [22]. Define ï£± ï£² min{m : m âˆˆ N, n | m(m + 1)}, if 2 | n, Z0 (n) = ï£³ min{m : m âˆˆ N, n | m2 }, if 2 - n. For any real x > 1, we have X

Z0 (2n âˆ’ 1) =

nâ‰¤x

3 3Î¶(3) 2 x + O x 2 + . Ï€2

X. Li [12]. Define

a(a + 1) C(n) = min a + b : a, b âˆˆ N, n | +b . 2 For any real x > 1, we have X

âˆš

2x 2 + O (x) ,

C(n)

1 C(n)

ln 2 Â·

d(C(n))

3 1 5 3 x ln x + x 2Î³ + ln 2 âˆ’ + O x4 , 2 2 2

nâ‰¤x

X nâ‰¤x

âˆš

2x + O (ln x) ,

where Î³ is the Euler constant. Y. Li [13]. Define

a(a + 1) D(n) = max ab : a, b âˆˆ N, n = +b . 2 For any real x > 1, we have X

D(n)

X C(n) D(n)

nâ‰¤x

âˆš 4 6 5 x 2 + O x2 , 45 âˆš 9 3 ln x + O (1) . 4

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No. 1

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[16] Yuanbing Lou. On the pseudo Smarandache function. Scientia Magna 3 (2007), no. 4, 48C50. [17] Weiyang Lu, Li Gao, Hongfei Hao and Xihan Wang. A lower bound estimate for the pseudo-Smarandache function. Journal of Shaanxi University of Science and Technology 32 (2014), no. 6, 180 - 183. (In Chinese with English abstract). [18] Abdullah-Al-Kafi Majumdar. A note on the pseudo-Smarandache function. Scientia Magna 2 (2006), no. 3, 1C25. [19] Abdullah-Al-Kafi Majumdar. A note on the near pseudo Smarandache function. Scientia Magna 4 (2008), no. 4, 104C111. [20] Richard Pinch. Some properties of the pseudo-Smarandache function. Scientia Magna 1 (2005), no. 2, 167 - 172. [21] JÂ´ ozsef SÂ´ andor. On a dual of the pseudo-Smarandache function. Smarandache Notions Journal 13 (2002), no. 1-3, 18C23. [22] Minna Tong. A new pseudo-Smarandache function and its mean value. Basic Sciences Journal of Textile Universities 26 (2013), no. 1, 18 - 20. (In Chinese with English abstract). [23] Xihan Wang, Li Gao and Weiyang Lu. A hybrid mean value of the pseudo-Smarandache function. Natural Science Journal of Hainan University 33 (2015), no. 2, 97 - 99. (In Chinese with English abstract). [24] Xihan Wang, Li Gao and Weiyang Lu. The hybrid mean value of the pseudo-Smarandache function. Henan Science 33 (2015), no. 10, 1682 - 1685. (In Chinese with English abstract). [25] Yu Wang. Some identities involving the near pseudo Smarandache function. Scientia Magna 3 (2007), no. 2, 44C49. [26] Hai Yang and Ruiqin Fu. On the mean value of the near pseudo Smarandache function. Scientia Magna 2 (2006), no. 2, 35C39. [27] Mingshun Yang. On a problem of the pseudo Smarandache function. Pure and Applied Mathematics 24 (2008), no. 3, 449 - 451. (In Chinese with English abstract). [28] Yongfeng Zhang. On the near pseudo Smarandache function. Scientia Magna 3 (2007), no. 1, 98C101. [29] Yani Zheng. On the pseudo Smarandache function and its two conjectures. Scientia Magna 3 (2007), no. 4, 74C76. [30] Xiaoyan Zhu and Li Gao. An equation involving Smarandache function. Journal of Yanan University (Natural Science Edition) 28 (2009), no. 2, 5 - 6. (In Chinese with English abstract).