ON SMARANDACHE TRIPLE FACTORIAL FUNCTION
You Qiying College of Science, Chang’an University, Xi’an, Shaanxi, P.R.China
Abstract
For any positive integer n, the Smarandache triple factorial function d3f (n) is defined to be the smallest integer such that d3f (n)!!! is a multiple of n. In this paper, we study the hybrid mean value of the Smarandache triple factorial function and the Mangoldt function, and give a sharp asymptotic formula.
Keywords:
Triple factorial numbers; Hybrid mean value; Asymptotic formula.
§1. Introduction According to [1], for any positive integer n, the Smarandache triple factorial function d3f (n) is defined to be the smallest integer such that d3f (n)!!! is a multiple of n. About this problem, we know very little. The problem is interesting because it can help us to calculate the Smarandache function. In this paper, we study the hybrid mean value of the Smarandache triple factorial function and the Mangoldt function, and give a sharp asymptotic formula. That is, we shall prove the following theorems. Theorem 1. If x ≥ 2, then for any positive integer k we have X
à 2
Λ1 (n)d3f (n) = x
n≤x
where
½
Λ1 (n) =
X am 1 k−1 + 2 m=1 logm x
log p, 0,
!
Ã
+O
!
x2 , logk x
if n is a prime p ; otherwise,
and am (m = 1, 2, · · · , k − 1) are computable constants. Theorem 2. If x ≥ 2, then for any positive integer k we have X n≤x
à 2
Λ(n)d3f (n) = x
X am 1 k−1 + 2 m=1 logm x
where Λ(n) is the Mangoldt function.
!
Ã
+O
!
x2 , logk x
90
SCIENTIA MAGNA VOL.1, NO.1
§2. One lemma To complete the proofs of the theorems, we need the following lemma. Lemma. For any positive integer α, if p ≥ (3α − 2) we have d3f (pα ) = (3α − 2)p.
Proof. Since [(3α − 2)p]!!! = (3α − 2)p · · · (3α − 3)p · · · p, so pα | [(3α − 2)p]!!!. On the other hand, if p ≥ (3α − 2), then (3α − 2)p is the smallest integer such that [(3α − 2)p]!!! is a multiple of pα . Therefore d3f (pα ) = (3α − 2)p.
§3. Proof of the theorems In this section, we complete the proofs of the theorems. Let ½
1, 0,
a(n) =
if n is prime; otherwise.
then for any positive integer k we have Ã
k−1 X m! x a(n) = π(x) = 1+ log x logm x m=1 n≤x
X
!
Ã
+O
x logk+1 x
By Abel’s identity we have X
Λ1 (n)d3f (n)
n≤x
=
X
p log p =
p≤x
X Z
= π(x) · x log x − Ã
= x
Z 2
π(t) (log t + 1) dt !
Ã
+O
Ã
Ã
Ã
= x
k−1 X
x
+O 2
2
x
m! 1+ m log x m=1
2
−
a(n)n log n
n≤x
x2 logk x
!
k−1 X m! X m! t t k−1 t+ +t m + log t log t log t m=1 logm t m=1
t (log t + 1) logk+1 t
!!
X am 1 k−1 + 2 m=1 logm x
dt !
Ã
+O
!
x2 , logk x
!
.
91
On Smarandache triple factorial function
where am (m = 1, 2, · · · , k − 1) are computable constants. Therefore X
Ã
p log p = x2
p≤x
X am 1 k−1 + 2 m=1 logm x
!
Ã
+O
!
x2 . logk x
So we have X
Ã
Λ1 (n)d3f (n) = x2
n≤x
X am 1 k−1 + 2 m=1 logm x
!
Ã
+O
!
x2 . logk x
This proves Theorem 1. It is obvious that d3f (pα ) ≤ (3α − 2)p. From Lemma 1 we have X
Λ(n)d3f (n)
n≤x
X
=
X
log p [(3α − 2)p] +
pα ≤x
log p [d3f (pα ) − (3α − 2)p] .
pα ≤x p<(3α−2)
Note that X
(3α − 2)p log p −
pα ≤x
=
X
x α≤ log log p
p≤x1/α
p log p(2α − 1) −
X
X
p log p
p≤x
X
=
X
X
p log p
pα ≤x
p log p(3α − 2)
x p≤x1/α 2≤α≤ log log p
X
¿
αx2/α log x1/α ¿ x log3 x
x 2≤α≤ log log p
and X pα ≤x p<(3α−2)
¿
X
log p [df (pα ) − (3α − 2)p] ¿
X
αp log p
x p<(3α−2) α≤ log log 2
X
(3α − 2)2 α log(3α − 2) ¿ log3 x,
x α≤ log log 2
so we have X n≤x
à 2
Λ(n)d3f (n) = x
X am 1 k−1 + 2 m=1 logm x
This completes the proof of Theorem 2.
!
Ã
+O
!
x2 . logk x
92
SCIENTIA MAGNA VOL.1, NO.1
References [1] “Smarandache k-factorial” at http://www. gallup. unm. edu/ smarandache/SKF.htm