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

2017·0.22 MB·

by  Huaning Liu

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
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