A survey on Smarandache notions in number theory II: pseudo-smarandache function

Transcription

1 Scientia Magna Vol. 07), No., A survey on Smarandache notions in number theory II: pseudo-smarandache function Huaning Liu School of Mathematics, Northwest University Xi an 707, China 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. 00 Mathematics Subject Classification A07, B50, L0, N5.. Definition and simple properties According to [], the pseudo-smarandache function is defined by mm + ) = min m : n. Some elementary properties can be found in [] and []. R. Pinch [0]. For any given L > 0 there are infinitely many values of n such that Zn ) > L, and there are infinitely many values of n such that > L. Zn + ) For any integer k, the equation The ration is not bounded. n = k has infinitely many solutions n. Fi < β < and integer t 5. The number of integers n with et < n < e t such that < n β is at most 96t e βt. The series is convergent for any α >. α Some eplicit epressions of for some particular cases of n were given by Abdullah- Al-Kafi Majumdar. A. A. K. Majumdar [8]. If p 5 is a prime, then Zp) = Z3p) = p, if 4 p, p, if 4 p +, p, if 3 p, p, if 3 p +,

2 46 H. Liu No. Z4p) = Z6p) = p, if 8 p, p, if 8 p +, 3p, if 8 3p +, 3p, if 8 3p +, p, if p, p, if p +, p, if 4 3p +, p, if 4 3p. A. A. K. Majumdar [8]. If p 7 is a prime, then p, if 0 p, p, if 0 p +, Z5p) = p, if 5 p, p, if 5 p +. If p is a prime, then p, if 7 p, Z7p) = p, if 7 p +, p, if 7 p, p, if 5 p +, 3p, if 7 3p, 3p, if 7 3p +. If p 3 is a prime, then p, if p, p, if p +, p, if p, Zp) = p, if p +, 3p, if 3p, 3p, if 3p +, 4p, if 4p, 4p, if 4p +, 5p, if 5p, 5p, if 5p +.

3 Vol. A survey on Smarandache notions in number theory II: pseudo-smarandache function 47 A. A. K. Majumdar [8]. Let p and q be two primes with q > p 5. Then Zpq) = min qy 0, p 0, where y 0 = min y :, y N, qy p =, 0 = min :, y N, p qy =. A. A. K. Majumdar [8]. If p 3 is a prime, then Zp ) = p. If p 5 is a prime, then Z3p ) = p. If p 3 is a prime and k 3 is an integer, then Zp k p k, if 4 p and k is odd, ) = p k, otherwise, Z3p k p k, if 3 p + and k is odd, ) = p k, otherwise. S. Gou and J. Li []. The equation = Zn + ) has no positive integer solutions. For any given positive integer M, there eists a positive integer s such that Zs) Zs + ) > M. Y. Zheng [9]. For any given positive integer M, there are infinitely many positive integers n such that Zn + ) > M. M. Yang [7]. Suppose that n has primitive roots. Then is a primitive root modulo n if and only if n =, 3, 4. W. Lu, L. Gao, H. Hao and X. Wang [7]. Let p 7 be a prime. Then we have Z p + ) 0p, Z p ) 0p. L. Gao, H. Hao and W. Lu [?]. Let p 7 be a prime, and let a, b be distinct positive integers. Then we have Z a p + b p ) 0p. Y. Ji [0]. Let r be a positive integer. Suppose that r,, 3, 5. Then Z r + ) + ) r

4 48 H. Liu No. Assume that r,, 4,. Then Z r ) + ) r Mean values of the pseudo-smarandache function Y. Lou [6]. For any real > we have ln = ln + O). W. Huang [9]. For any integer n > we have n k= ln Zk) ln k n ) = + O, ln n ) ln Zk) = O. ln n k n L. Cheng [4]. Let pn) denote the smallest prime divisor of n, and let k be any fied positive integer. For any real > we have pn) = k ln + i= ) a i ln i + O ln k+, where a i i =, 3,, k) are computable constants. X. Wang, L. Gao and W. Lu [3]. Define 0, if n =, Ωn) = α p + α p + + α r p r, if n = p α pα pαr r. Let k be any fied positive integer. For any real > we have k Ωn) = ζ3)3 3 ln + a i 3 ) 3 ln i + O ln k+, where a i i =, 3,, k) are computable constants. H. Hao, L. Gao and W. Lu [8]. Let dn) denote the divisor function, and let k be any fied positive integer. For any real > we have dn) = π4 36 i= k ln + i= where a i i =, 3,, k) are computable constants. X. Wang, L. Gao and W. Lu [4]. Define Dn) = min m : m N, n a i ) ln i + O ln k+, m di). i=

5 Vol. A survey on Smarandache notions in number theory II: pseudo-smarandache function 49 Let k be any fied positive integer. For any real > we have ln Dn) = ζ3) ln 3 where a i i =, 3,, k) are computable constants. 3 k ln + a i 3 ) 3 ln i + O ln k+, 3. The dual of the pseudo-smarandache function, the near pseudo-smarandache function, and other generalizations According to [], the dual of the pseudo-smarandache function is defined by Z n) = ma m N : i= mm + ) n. D. Liu and C. Yang [5]. Let A denote the set of simple numbers. For any real we have Z n) = C ln + C ) ln + O ln 3, where C, C are computable constants. X. Zhu and L. Gao [30]. We have Z n) n α = ζα) m= m m α m + ) α. The near pseudo Smarandache function Kn) is defined as where kn) = min derived in [9]. k : k N, n Kn) = n i + kn), i= n i + k. Some recurrence formulas satisfied by Kn) were i= H. Yang and R. Fu [6]. For any real we have d Kn) φ Kn) ) nn + ) ) nn + ) = 3 ) 4 log + A + O log, = 93 8π + O 3 +ɛ), where φn) denotes the Euler function, A is a computable constant, and ɛ > 0 is any real number. Y. Zhang [8]. For any real number s >, the series K s n)

6 50 H. Liu No. is convergent, and Kn) = 3 ln + 5 6, K n) = 08 π + ln. 7 Y. Li, R. Fu and X. Li [4]. We have ) Kn) = ln ln 3 ln + B ln + ln ln 9 ln + O ln, Kn) = 3 ln ln ) + D ln ln + E + O ) ln ln. ln L. Gao, R. Xie and Q. Zhao [5]. Define p d n) = d n d, q d n) = d n d<n d. For any real > we have K p d n)) = K q d n)) = ln ln ln + A ln + 5 ln ln ln ln + A ln + 9 ln ) 5 ln ln + O ln, ) 3 ln ln + O ln, where A, A are computable constants. Other generalizations on the near pseudo-smarandache function have been given. eample, define mm + )m + ) Z 3 n) = min m : m N, n. 6 The elementary properties were studied in [6] and [7]. Y. Wang [5]. Define For U t n) = min k : t + t + + n t + k = m, n m, k, t, m N. For any real number s >, we have U sn) = ζs) ) s, U s n) = ζs) U s 3 n) = ζs) + 5 s 6 s + s ) 3 s )), + ) ) s.

7 Vol. A survey on Smarandache notions in number theory II: pseudo-smarandache function 5 M. Tong []. Define minm : m N, n mm + ), if n, Z 0 n) = minm : m N, n m, if n. For any real >, we have Z 0 n ) = 3ζ3) π + O 3 +ɛ). X. Li []. Define aa + ) Cn) = min a + b : a, b N, n + b. For any real >, we have Cn) = 3 + O ), Cn) where γ is the Euler constant. = ln + O ln ), dcn)) = ln + γ + 5 ln 3 ) ) + O 3 4, Y. Li [3]. Define aa + ) Dn) = ma ab : a, b N, n = + b. For any real >, we have Dn) = O ), Cn) Dn) = ln + O ). References [] Charles Ashbacher. Pluckings from the tree of Smarandache sequences and functions. American Research Press ARP), Lupton, AZ, 998. [] Su Gou and Jianghua Li. On the pseudo-smarandache function. Scientia Magna 3 007), no. 4, 8C83.

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