Some Properties On Arithmetic Functions

There are a lot of investigations on arithmetic functions: and , where (n) is Euler totient function and o (n) is the sum of divisors of n. In this thesis, we recall some research on properties of Arithmetic functions, for example, the composite integers n with n (n)=2(mod (n)), Lehmer conjecture and some results on multiply number.We investigate the solutions of n o (n)=in(mod (n))(4 rm), and positive integersn for which (n) is a divisor of o (n). The following results are proved: Theoreml. Let n be a positive integer and n>2 withwhere is a positive integer (i-1,. . .,s), pi,. . .,ps are distinct odd primes. Then n has the form:Especially, untrivial solutions of n (n) = P (mod (n)) (n > 2) must be, where p is an odd prime, / i s a positive integer, (n=l, 2 arecalled trivial solutions).Theorem 2 . Let n be a positive integer and n>2 withwhere l1 isa positive integer (i=1,......,s), p1,. . .,ps are distinct odd primes. Then n has the form:where q is an odd prime,IVTheorem 3. The positive integers n with o> (n) =2 and (n)| o (n) are only 6,12, 56,15,35,2?(2?2-l) (where 2 ?2-l are prime and a eN) .Theorem4(a) The positive integers n with w (n ) =3 ,2 ^ n and <1> (n)| o (n) are only3-5-7, 5- 11-19, 33-5-ll, 33 ?11-17, 3?7- (2-3+1-l) (where 2 ?304"1 - 1 areprimeand a eN) .(b) The positive integers n with w (n) =3, 2|nand o (n)=k (n)(where k 5 and ke N) are only (where 2 ?2-l are prime, a eN) .(c) There are not integers n with w (n) =3, 2 |n,n?t2皃q (a eN) and o (n)= k 4> (n)(where k =$4 and k e N, p, q are distinct odd primes).It's worth noting 4> (n)| o (n) has solutions: 2" (2?*2- 1), which has the similar form as the even perfect numbers 2" '(2n- 1) (where 2?*2- 1 and 2n- 1 are prime, a eN, neN).