SCIENTIA MAGNA, Vol. 16, No. 1, 2021

2021·3.5 MB·English

Vol. 16, No. 1, 2021 ISSN 1556-6706

SCIENTIA MAGNA
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School of Mathematics
Northwest University
Xi’an, Shaanxi, China
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i
Contents
Yuchan Qi: A survey on Smarandache notions in number theory: proper divisor product,
sieve sequences, integer part sequences, pseudo-odd, pseudo-even and related sequences 1
Zehua Liu: A survey on Smarandache notions in number theory: Pseudo-Smarandache-
Squarefree function and Smarandache dual function 35
Ahmed Bachir, T. Prasad and Nawal Ali Sayyaf: Asymmetric Fuglede - Putnam
theorem for p−(α,β)-normal operators 51
Lakshika Chutani, Garima Manocha and Niraj Kumar: On Γ-ring for a class of
Entire Dirichlet series in two variables 59
(e)
Yang Li: A short interval result for the function τ (n) with a negative r-th power 67
3
Senol Dost: Minimal and maximal open sets in ditopological texture space 72
(cid:16) (cid:17)2
Fuxia Xu: The mean value for the function t(e)(n) over square-full numbers 79
Rasul Rasuli: Norms over intuitionistic fuzzy SU-subalgebras 84
(cid:16) (e) (cid:17)r
Ruiyang Yue: The mean value of function q (n) over cube-full number 97
k
G. Sharath: On some Somos’s η-function identities of level 27 and their partition inter-
pretations 103
Lu Zhang: A short interval for the function (φ(e)(n))r 110
ii
Scientia Magna
Vol. 16 (2021), No. 1, 1-34
A survey on Smarandache notions in number
theory: proper divisor product, sieve sequences,
integer part sequences, pseudo-odd,
pseudo-even and related sequences
Yuchan Qi
School of Mathematics, Northwest University
Xi’an 710127, China
E-mail: [email protected]
Abstract InthispaperwegiveasurveyonSmarandachedivisorproductsequences,sievese-
quences,integerpartsequences,Smarandachepseudo-odd,pseudo-evenandpseudo-multiples
sequences, Smarandache kn-digital sequences and related sequences.
Keywords proper divisor product, sieve sequences, integer part, pseudo-odd, pseudo-even,
pseudo-multiples, kn-digital.
2010 Mathematics Subject Classification 11B83, 11N25, 11N37.
§1. Introduction and preliminaries
Number Theory is one of the oldest of mathematical disciplines. Problems are often easy
tostate,butextremelydifficulttosolve,whichistheoriginofmuchoftheircharm. Peoplehave
always been interested in the ”purity” of the integers. Divisibility is the backbone of number
theory, which has a close relationship with prime factorizations, therefore the values of number
theoretic functions can often be computed by formulas based on the prime factorization. For
example: The Euler phi function Φ(n) is the number of integers m, where 1 ≤ m ≤ n and m
and n are relatively prime. Sum of divisors function σ(n) is the sum of all the positive divisors
of n.
Inthe1970’s, FlorentinSmarandachecreatedanewfunctioninnumbertheory. LetS(n),
for n ∈ N+ denote the Smarandache function, then S(n) is defined as the smallest m ∈ N+,
with n | m!. The consequences of this simple definition encompass many areas of mathe-
matics. This paper is a survey on Smarandache divisor product sequences, Smarandache sieve
sequences,integerpartsequences,LCMratiosequences,Smarandachepseudo-odd,pseudo-even
and pseudo-multiples sequences, simple sequences, primitive numbers, Smarandache kn-digital
sequences.
2 Y.Qi No. 1
§2. Smarandache divisor product sequences
F.Smarandache introduced the function P (n) := (cid:81)d in Problem 25[1]. For example,
d
d|n
P (1)=1,P (2)=2,P (3)=3 P (4)=8,··· ,P (p)=p,···. q (n) denotes the product of all
d d d d d d
(cid:81)
proper divisors of n. That is, q (n) = d. For example, q (1) = 1,q (2) = 1,q (3) =
d d|n,d<n d d d
1,q (4)=2,··· . In problem 25 and 26 of [1], Professor F. Smarandache asked people to study
d
the properties of the sequences {p (n)} and {q (n)}. F. Liang studied the properties of the
d d
sequences of {p (n)} and {q (n)} for k -th divisor sum function and gave two more general
d d
results.
F. Liang [28] Let n = pα,p be a prime and α be a positive integer. Then for any fixed
positive integer k, we have the inequality
1
σ (φ(p (n)))≥ pk(n),
k d 2k d
1
σ (φ(q (n)))≥ qk(n),
k d 2k d
where σ (n)=(cid:80) dk is the k -th divisor sum function.
k d|n
A similar function P (n) denotes the product of all square-free divisors of n, i.e.,
sd
(cid:89)
P (n):= d.
sd
d|n
µ(d)(cid:54)=0
Y. Han [15] We have the asymptotic formula
(cid:88)logPsd(n)=A1xlog2x+A2xlogx+A3x+O(cid:16)x12 exp(cid:16)−D(logx)35(loglogx)−51(cid:17)
n≤x
where A ,A ,A are constants, D >0 is an absolute constant.
1 2 3
(cid:81)
Let 1≤l≤q be fixed integers. Define a function P (n):= d.
ld
d|n
d≡l(modq)
X. Ma [46] For any real number x>1, we have
(cid:88)logPad(n)= 21qxlog2x+ γ−q 1xlogx+cx+O((q−1x)2872+ε),
n≤x
where c is a constant which depends on q and l, γ is the Euler constant, ε is a fixed small
positive constant.
For any positive integer n ≥ 1, the Smarandache Superior Prime Part P (n) is defined as
p
thesmallestprimenumbergreaterthanorequalton. Forexample,thefirstfewvaluesofP (n)
p
are P (1) = 2,P (2) = 2,P (3) = 3,P (4) = 5,P (5) = 5,P (6) = 7,P (7) = 7,P (8) = 11
p p p p p p p P
P (9) = 11,P (10) = 11,P (11) = 11,P (12) = 13,P (13) = 13,P (14) = 17,P (15) = 17 ···
p p p p p p p
Foranypositiveintegern≥2, definetheSmarandacheInferiorPrimePartp (n)asthelargest
p
prime number less than or equal to n. Its first few values are p (2) = 2 p (3) = 3,p (4) =
p p p
3,p (5) = 5,p (6) = 5,p (7) = 7,p (8) = 7,p (9) = 7,p (10) = 7 p (11) = 11,···. There is
p p p p p p p
a close relationship between the Smarandache prime part and the prime distribution problem.
Define
I ={p (2)+p (3)+···+p (n)}/n,
n p p p
2
Vol. 16 AsurveyonSmarandachenotionsinnumbertheory 3
and
S ={P (2)+P (3)+···+P (n)}/n.
n p p p
There are two problems in problem 10 of reference [22]:
(A). If lim (S −I ) converges or diverges. If it converges, find the limit.
n n
n→∞
(B). If lim Sn converges or diverges. If it converges, find the limit.
n→∞ In
The problem (A) is still open. But for the problem (B), X. Yan got asymptotic properties
of Sn, and gave a shaper asymptotic formula for it.
In
X. Yan [65] For any positive integer n>1, we have the asymptotic formula
S (cid:16) (cid:17)
n =1+O n−31 .
I
n
From this theorem it can be deduced that the limit S /I converges as n−→∞, and
n n
S
lim n =1.
n→∞ In
This solved the problem B of reference [22].
For any positive integer n, a new additive function F(n) is defined as F(0) = 0 and
F(n)=α p +α p +···+α p for n=pα1pα2···pαk.
1 1 2 2 k k 1 2 k
S. Gou [10] Let N be a positive integer. Then for any fixed real number x>1, we have
(cid:88) (cid:88)N x2 (cid:18) x2 (cid:19)
F (P (n))= d · +O ,
d i lnix lnN+1x
n∈x i=1
where d ,··· ,d ) are computable constants and u = π4. And we also have
1 N 1 72
(cid:88) (cid:88)N x2 (cid:18) x2 (cid:19)
F (q (n))= h · +O ,
d i lnix lnN+1x
n∈x i=1
where h ,··· ,h ) are computable constants and h = π4 − π2.
1 N 1 72 12
T. Zhang [82] For any real number x≥1, we have the asymptotic formula
(cid:88)lnPd(n)= 12xln2x+(C−1)xlnx−(C−1)x+O(cid:16)x12 lnx(cid:17),
n≤x
where C is the Euler constant.
For any real number x≥1, we have the asymptotic formula
(cid:88)lnqd(n)=xln2x+(C−2)xlnx−(C−2)x+O(cid:16)x12 lnx(cid:17).
n≤x
W. Zhu [84] For any real number x>1, we have the asumptotic formula
(cid:18) (cid:19)
(cid:88) 1 1
=lnlnx+C +O ,
P (n) 1 lnx
d
n≤x
3
4 Y.Qi No. 1
where C is a constant. And we also have
1
(cid:18) (cid:19)
(cid:88) 1 lnlnx
=π(x)+(lnlnx)2+Blnlnx+C +O ,
q (n) 2 lnx
d
n≤x
where π(x) is the number of all primes ≤x,B and C are constants.
2
H. Liu and W. Zhang gave the following results.
H. Liu [33] For any positive integer n, we have the inequality
1
σ(φ(P (n)))≥ P (n),
d 2 d
where φ(k) is the Euler’s function and σ(k) is the divisor sum function. And we also have the
inequality
1
σ(φ(q (n)))≥ η (n).
d 2 d
M. Le gave a formulae of P (n).
d
M. Le [27] Let n=pr1...prk be the factorization of n and let
1 k

 1/2(r +1)...(r +1) if n is not a square,
1 k
r(n)=
 1/2((r1+1)...(rk+1)−1) if n is a square.
Then we have P (n)=nr(n).
d
§3. Smarandache sieve sequences
The definition of Smarandache irrational root sieve is: from the set of natural numbers
(except 0 and 1 ): take off all powers of 2k,k ≥ 2; take off all powers of 3k,k ≥ 2; take off
all powers of 5k,k ≥ 2; take off all powers of 6k,k ≥ 2; take off all powers of 7k,k ≥ 2; take
off all powers of 10k,k ≥ 2,··· (take off all k -powers, k ≥ 2 ). For example: 2,3,5,6,7,10,
11,12,13,14,15,17,18,19··· are all irrational root sieve sequence. Let A denotes the set of all
the irrational root sieve. X. Zhang studied the mean value of the irrational root sieve sequence
and gave an interesting asymptotic formula for it.
X. Zhang and Y. Lou [78] Let d(n) denote the divisor function. Then for any real
mumber x≥1, we have the asymptotic formula
(cid:88)d(n) = (cid:18)x− 4π32√xlnx+A1x13 ln2x+A2x31 lnx+A3x31 +A4√x(cid:19)lnx
n∈A
√ (cid:16) (cid:17)
+(2γ−1)x+A5 x+A6x13 +O x143299+(cid:15) ,
where (cid:15) denotes any fixed positive number, γ is the Euler constant, A ,A ,A A ,A ,A are
1 2 3 4 5 6
the computable constants.
Let A denote the set of all numbers in the k -th power free sieve sequence.
4
Vol. 16 AsurveyonSmarandachenotionsinnumbertheory 5
J. Su [51] Let k ≥ 2 be a fixed positive integer, A denotes the set of all power k sieve
sequence. Then for any real x>1, we have the asymptotic formula
(cid:88)1=c(k)·x+O(cid:16)xk1(cid:17),
n≤x
n∈A
where c(k)=(cid:81)∞ (cid:0)1− 1 (cid:1) is a constant, and c(2)= 1. Foranyrealnumberx>1,fromthe
n=2 nk 2
Prime Number Theorem we know that there are at most O(cid:0) x (cid:1) primes in the interval [1,x],
lnx
so from the theorem above one can get that there are an infinity of numbers of the power k
sieve sequence which are not prime. Therefore, there is an infinity of numbers of the power k
sieve sequence which are not prime.
J. Guo and X. Zhao studied the convergent property of some infinite series involving this
sequence and gave some interesting identities.
J. Guo and X. Zhao [14] 1. Let k ≥ 2 be any positive integer. For any real number
α>1, we have the identity:
∞
(cid:88) 1 ζ(α)
= ,
nα ζ(kα)
n=1
n∈A
where ζ(s) denotes the Riemann-zeta function. Moreover, let B be the set of all numbers in the
square free sieve seaquence C be the set of all numbers in the cubic free sieve sequence. Then
∞ ∞
(cid:88) 1 15 (cid:88) 1 315
= and = .
n2 π2 n2 2π4
n=1 n=1
n∈B n∈C
2. Let k ≥2 be any positive integer. For any real number α>1, we have the identity:
(cid:88)∞ d(n) ζ2(α) (cid:89)(cid:18) k(pα−1) (cid:19)
= 1− .
nα ζ(kα) p(k+1)α−pα
n=1 p
n∈A
X. Pan and B. Liu [48] Let A denote the set of all elements of the irrational root sieve
sequence. Then for any real number x>1, we have the asymptotic formula
(cid:88)1=x− 6 √x− 6 x13 +O(cid:16)x14 ·lnx(cid:17).
π2 π2
a≤x
a∈A
The odd sieve sequence is the sequence which is composed of all odd number that are not
equal to the difference of two primes. For example: 7,11,19,23,25,···. Let A denote the set
of the odd sieve numbers. W. Yao used analytic method to study the mean value properties of
this sequence and gave two interesting asymptotic formulae.
W. Yao [63] 1. For any positive number x>1, we have the asymptotic formula
(cid:88) x2 x2 (cid:18) x2 (cid:19)
n= − +O .
4 2lnx ln2x
n≤x
n∈A
5
6 Y.Qi No. 1
2. For any positive number x>1, we have the asymptotic formula
(cid:18) (cid:19)
(cid:88) 1 1 x 1 1
= ln −lnln(x+2)+ γ−A+B+O ,
n 2 2 2 lnx
n≤x
n∈A
where A,B are computable constants, γ is the Euler’s constant.
R. Xie [61] Let ϕ be Euler function. For any positive number x > 1, we have the
asymptotic formula
(cid:88)ϕ(n)= 3 +O(cid:16)x32+(cid:15)(cid:17),
π2
n≤x
n∈A
where (cid:15) is any fixed positive number.
(cid:80)
L. Qian [50] Let σ(n) denote the sum of positive factor of n: σ(n) = d. Then for
d|n
any real number x(cid:62)1 we have
(cid:88)σα(n)= 21ζ(α+1)g(α+1)x2+O(cid:16)x23+ε(cid:17),
n≤x
n∈A
where g(α)=(cid:81)(1+(p−1)p−s+(1−p)pα−2s), ζ(s) is Riemann zeta- function and ε is a fixed
p
positive number.
Let p be a prime, e (n) denote the largest exponent of power p which is included n. L.
p
Qi studied the mean value properties of e (n) acting on the irrational root sieve sequences and
p
gave an asymptotic formula.
L. Qi [49] Let A denote the set of all elements of the irrational root sieve sequence. For
any real number x(cid:62)1, we have
(cid:88)ep(n)= p−1 1x− p−2 1x21 − p−3 1x31 +O(cid:16)x12+ε(cid:17).
n(cid:54)x
n∈A
§4. The Inferior and Superior parts of integers sequences
For any positive integer n, the Smarandache Superior m-th power part sequence SSMP(n)
is defined as the smallest m -th power greater than or equal to n. The Smarandache Inferior
m -th power part sequence SIMP(n) is defined as the largest m -th power less than or equal to
n. For example, if m = 2, then the first few terms of SIMP(n) are: 0,1,1,1,4,4,4,4,4,9,···.
The first few terms of SSMP(n) are 1,4,4,4,9,9,9,9,9,.16,16,···. Now we let
S =(SSMP(1)+SSMP(2)+···+SSMP(n))/n
n
I =(SIMP(1)+SIMP(2)+···+SIMP(n))/n
n
(cid:112)
K = n SSMP(1)+SSMP(2)+···+SSMP(n)
n
(cid:112)
I = n SIMP(1)+SIMP(2)+···+SIMP(n).
n
6