An Inequality Concerning the Smarandache Function

2 Pages·1998·0.08 MB·

by  Maohua Le

Smarandache Notions Journal, Vol. 9, No.1-2-3, 1998, pp. 124-125.
Maoh:.::.a Le
Zhanjiang Normal College, Zhanjang, Guangdong, ?R.China
Abstract. For any positive integer n, let S(n) denote the
S~arandache function of n. In this paper Ne prove that
S(m)+S(n).
S(~n)~
Let N be the set of a~~ posicive integers. For any posicive
integer n, let S(n) denote the Smarandache function of n.
By ~2~, '.f'le ::ave
: -: \ S(nj= min{~!kE N, Dik!}.
\ - I
~ecent:y, Jozsef[l] proved that
(2) S(mn)~ rnS(n), m,n E N.
In this paper we give a considerable improvement for the ".lpper
bound (2). We prove the following result.
Theorem. For any positive integers m,n, we have
S(m)+S(n).
S(rnn)~
Proof. a=S(m) and b=5(n) Then we have
~et
,.,Ib'
l ... j :,
~et x be a positive integer with x ~ 0, and :et
x (x-I)
a!
x
be a binomial coeficient. It is a wel: k~own fact that a
a positive integer. 50 we r-:ave
~s
1,J-;' a ;' :: _x (x-1.J. . I\ ••• (\ x-a·T 1_ \/ ,
by ! 4- \ . ::..:rther, since m!a!, we get from (5)
, /
(6 ; mix(x-1) ... (x-a+l),
fer any positive integer x Nith x ~ a. ?u~ x=a+b. We see
from (3) and (6) that
(i \
I. ! i
?ererences
1. S.Jozsef,On certain inegualites involving t~e Smarandac~e
function, Smarandache No~ce J. 7(l996), No.1-3, 3-6.
2. ? Smarandache "A function in the nurrber t.heory"," Smarandac:-.e
fU:1ction J". 1(1990), No.1, 3-17.
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