Some identities involving the near pseudo Smarandache function

2007·0.17 MB·

by  Yu Wang

Scientia Magna
Vol. 3 (2007), No. 2, 44-49
Some identities involving the near pseudo
Smarandache function
Yu Wang
Department of Mathematics, Northwest University
Xi’an, Shaanxi, P.R.China
Received March 29, 2007
Abstract For any positive integer n and fixed integer t ≥ 1, we define function U (n) =
t
min{k : 1t+2t+···+nt+k = m, n | m, k ∈ N+, t ∈ N+}, where n ∈ N+, m ∈ N+,
which is a new pseudo Smarandache function. The main purpose of this paper is using the
elementary method to study the properties of U (n), and obtain some interesting identities
t
involving function U (n).
t
Keywords Some identities, reciprocal, pseudo Smarandache function.
§1. Introduction and results
In reference [1], A.W.Vyawahare defined the near pseudo Smarandache function K(n) as
n(n+1)
K(n)=m= +k,wherek isthesmallpositiveintegersuchthatndividesm. Thenhe
2
studied the elementary properties of K(n), and obtained a series interesting results for K(n).
n(n+3) n(n+2)
For example, he proved that K(n)= , if n is odd, and K(n)= , if n is even;
2 2
The equation K(n) = n has no positive integer solution. In reference [2], Zhang Yongfeng
studied the calculating problem of an infinite series involving the near pseudo Smarandache
1 (cid:88)∞ 1
functionK(n), and proved that for anyreal numbers> , the series is convergent,
2 Ks(n)
n=1
and
(cid:88)∞ 1 2 5
= ln2+ ,
K(n) 3 6
n=1
(cid:88)∞ 1 11 22+2ln2
= π2− .
K2(n) 108 27
n=1
Yang hai and Fu Ruiqin [3] studied the mean value properties of the near pseudo Smarandache
function K(n), and obtained two asymptotic formula by using the analytic method. They
proved that for any real number x≥1,
(cid:88) (cid:88) (cid:181) n(n+1)(cid:182) 3 (cid:179) (cid:180)
d(k)= d K(n)− = xlogx+Ax+O x12 log2x ,
2 4
n≤x n≤x
Vol. 3 SomeidentitiesofreciprocalofthenearpseudoSmarandachefunction 45
where A is a computable constant.
(cid:88) (cid:181) n(n+1)(cid:182) 93 (cid:179) (cid:180)
ϕ K(n)− = x2+O x32+(cid:178) ,
2 28π2
n≤x
where (cid:178) denotes any fixed positive number.
In this paper, we define a new near Smarandache function U (n)=min{k : 1t+2t+···+
t
nt+k =m, n|m, k ∈N+, t∈N+}, where n∈N+, m∈N+. Then we study its elementary
properties. Aboutthisfunction,itseemsthatnonehadstudiedityet,atleastwehavenotseen
such a paper before. In this paper, we using the elementary method to study the calculating
problem of the infinite series
(cid:88)∞ 1
,
Us(n)
n=1 t
and give some interesting identities. That is, we shall prove the following:
Theorem 1. For any real number s>1, we have the identity
(cid:181) (cid:182)
(cid:88)∞ 1 1
=ζ(s) 2− ,
Us(n) 2s
n=1 1
where ζ(s) is the Riemann zeta-function.
Theorem 2. For any real number s>1, we have
(cid:183) (cid:181) (cid:182)(cid:181) (cid:182)(cid:184)
(cid:88)∞ 1 1 1 1 1
=ζ(s) 1+ − +2 1− 1− .
Us(n) 5s 6s 2s 3s
n=1 2
Theorem 3. For any real number s>1, we also have
(cid:34) (cid:181) (cid:182) (cid:35)
(cid:88)∞ 1 1 2
=ζ(s) 1+ 1− .
Us(n) 2s
n=1 3
π2 π4
Taking s = 2, 4, and note that ζ(2) = , ζ(4) = , from our theorems we may
6 90
immediately deduce the following:
Corollary. Let U (n) defined as the above, then we have the identities
t
(cid:88)∞ 1 7 (cid:88)∞ 1 2111
= π2; = π2;
U2(n) 24 U2(n) 5400
n=1 1 n=1 2
(cid:88)∞ 1 25 (cid:88)∞ 1 31
= π2; = π4;
U2(n) 96 U4(n) 1440
n=1 3 n=1 1
t
(cid:88)∞ 1 2310671 (cid:88)∞ 1 481
= π4; = π4.
U4(n) 72900000 U4(n) 23040
n=1 2 n=1 3
46 YuWang No. 2
§2. Some lemmas
To complete the proof of the theorems, we need the following several lemmas.
Lemma 1. For any positive integer n, we have

n
 , if 2|n,
U (n)= 2
1 
n, if 2†n.
Proof. See reference [1].
Lemma 2. For any positive integer n, we also have

5
 6n, if n≡0(mod 6),
n, if n≡1(mod 6) or n≡5(mod 6),
U2(n)= nn2, if n≡2(mod 6) or n≡4(mod 6),
, if n≡3(mod 6).
3
Proof. It is clear that
U (n) = min{k :12+22+···+n2+k =m,n|m,k ∈N+}
2
n(n+1)(2n+1)
= min{k : +k ≡0(mod n),k ∈N+}.
6
(1) If n≡0(mod 6), then we have n=6h (h =1,2···),
1 1
n(n+1)(2n+1) 6h (6h +1)(12h +1)
= 1 1 1
6 6
= 72h3+18h2+h ,
1 1 1
n(n+1)(2n+1) 5n
so n| +U (n) if and only if 6h |h +U (n), then U (n)= .
6 2 1 1 2 2 6
(2) If n≡1(mod 6), then we have n=6h +1(h =0,1,2···),
2 2
n(n+1)(2n+1) (6h +1)(6h +2)(12h +3)
= 2 2 2
6 6
= 12h2(6h +1)+7h (6h +1)+6h +1,
2 2 2 2 2
n(n+1)(2n+1) n(n+1)(2n+1)
because n | , so n | +U (n) if and only if n | U (n), then
6 6 2 2
U (n)=n.
2
If n≡5(mod 6), then we have n=6h +5(h =0,1,2···),
2 2
n(n+1)(2n+1) (6h +5)(6h +6)(12h +11)
= 2 2 2
6 6
= 12h2(6h +5)+23h (6h +5)+11(6h +5),
2 2 2 2 2
n(n+1)(2n+1) n(n+1)(2n+1)
because n | , so n | +U (n) if and only if n | U (n), then
6 6 2 2
U (n)=n.
2
(3) If n≡2(mod 6), then we have n=6h +2(h =0,1,2···),
2 2
n(n+1)(2n+1) (6h +2)(6h +3)(12h +5)
= 2 2 2
6 6
= 12h2(6h +2)+11h (6h +2)+2(6h +2)+3h +1,
2 2 2 2 2 2
Vol. 3 SomeidentitiesofreciprocalofthenearpseudoSmarandachefunction 47
n(n+1)(2n+1) n
so n| +U (n) if and only if 6h +2|3h +1+U (n), then U (n)= .
6 2 2 2 2 2 2
If n≡4(mod 6), then we have n=6h +4(h =0,1,2···),
2 2
n(n+1)(2n+1) (6h +4)(6h +5)(12h +9)
= 2 2 2
6 6
= 12h2(6h +4)+19h (6h +4)+7(6h +4)+3h +3,
2 2 2 2 2 2
n(n+1)(2n+1) n
so n| +U (n) if and only if 2(3h +2)|3h +2+U (n), then U (n)= .
6 2 2 2 2 2 2
(4) If n≡3(mod 6), then we have n=6h +3(h =0,1,2···),
2 2
n(n+1)(2n+1) (6h +3)(6h +4)(12h +7)
= 2 2 2
6 6
= 12h2(6h +3)+15h (6h +3)+4(6h +3)+4h +2,
2 2 2 2 2 2
n(n+1)(2n+1) n
so n| +U (n) if and only if 3(2h +1)|2(2h +2)+U (n), then U (n)= .
6 2 2 2 2 2 3
Combining (1), (2), (3) and (4) we may immediately deduce Lemma 2.
Lemma 3. For any positive integer n, we have

n
 , if n≡2(mod 4),
U (n)= 2
3 
n, otherwise.
Proof. From the definition of U (n) we have
3
U (n) = min{k :13+23+···+n3+k =m,n|m,k ∈N+}
3
n2(n+1)2
= min{k : +k ≡0(mod n),k ∈N+}.
4
(a) If n≡2(mod 4), then we have n=4h +2(h =0,1,2···),
1 1
n2(n+1)2
=(4h +2)3(2h +1)+(4h +2)2(2h +1)+(2h +1)2,
4 1 1 1 1 1
n2(n+1)2 n
so n| if and only if 2(2h +1)|(2h +1)2+U (n), then U (n)= .
4 1 1 3 3 2
(b) If n≡0(mod 4), then we have n=4h (h =1,2···),
2 2
n2(n+1)2
=4h2(4h +1)2,
4 2 2
n2(n+1)2
so n| +U (n) if and only if n|U (n), then U (n)=n.
4 3 3 3
If n≡1(mod 4), then we have n=4h +1(h =0,1,2···),
1 1
n2(n+1)2
=(4h +1)2(2h +1)2,
4 1 1
n2(n+1)2
so n| +U (n) if and only if n|U (n), then U (n)=n.
4 3 3 3
If n≡3(mod 4), then we have n=4h +3(h =0,1,2···),
1 1
n2(n+1)2
=4(4h +3)2(h +1)2,
4 1 1
n2(n+1)2
uso n| +U (n) if and only if n|U (n), then U (n)=n.
4 3 3 3
Now Lemma 3 follows from (a) and (b).
48 YuWang No. 2
§3. Proof of the theorems
Inthissection,weshallusetheelementarymethodstocompletetheproofofthetheorems.
First we prove Theorem 1. For any real number s>1, from Lemma 1 we have
(cid:181) (cid:182)
(cid:88)∞ 1 (cid:88)∞ 1 (cid:88)∞ 1 (cid:88)∞ 1 (cid:88)∞ 1 1
= + = + =ζ(s) 2− ,
Us(n) (n)s ns hs (2h+1)s 2s
n=1 1 h=1 2 h=0 h=1 h=0
n=2h n=2h+1
where ζ(s) is the Riemann zeta-function. This proves Theorem 1.
For t=2 and real number s>1, from Lemma 2 we have
(cid:88)∞ 1 (cid:88)∞ 1 (cid:88)∞ 1 (cid:88)∞ 1 (cid:88)∞ 1 (cid:88)∞ 1
= (cid:161) (cid:162) + + (cid:161) (cid:162) + (cid:161) (cid:162) +
Us(n) 5n s ns n s n s ns
n=1 2 h1=1 6 h2=0 h2=0 3 h2=0 2 h2=0
n=6h1 n=6h2+1 n=6h2+2 n=6h2+4 n=6h2+5
(cid:88)∞ 1 (cid:88)∞ 1 (cid:88)∞ 1 (cid:88)∞ 1
= + + + +
(5h )s (6h +1)s (3h +1)s (2h +1)s
1 2 2 2
h1=1 h2=0 h2=0 h2=0
(cid:88)∞ 1 (cid:88)∞ 1
+
(3h +2)s (6h +5)s
2 2
h2=0(cid:183) h2=0(cid:181) (cid:182)(cid:181) (cid:182)(cid:184)
1 1 1 1
= ζ(s) 1+ − +2 1− 1− ,
5s 6s 2s 3s
This completes the proof of Theorem 2.
If t=3, then for any real number s>1, from Lemma 3 we have
(cid:88)∞ 1 (cid:88)∞ 1 (cid:88)∞ 1 (cid:88)∞ 1 (cid:88)∞ 1
= + + (cid:161) (cid:162) +
Us(n) ns ns n s ns
n=1 3 h2=1 h1=0 h1=0 2 h1=0
n=4h2 n=4h1+1 n=4h1+2 n=4h1+3
(cid:88)∞ 1 (cid:88)∞ 1 (cid:88)∞ 1 (cid:88)∞ 1
= + + +
(4h )s (4h +1)s (2h +1)s (4h +3)s
2 1 1 1
h2=1(cid:34) (cid:181) h1=0(cid:182) (cid:35) h1=0 h1=0
1 2
= ζ(s) 1+ 1− ,
2s
This completes the proof of Theorem 3.
Open Problem. For any integer t > 3 and real number s > 1, whether there exists a
calculating formula for the Dirichlet series
(cid:88)∞ 1
?
Us(n)
n=1 t
This is an open problem.
Vol. 3 SomeidentitiesofreciprocalofthenearpseudoSmarandachefunction 49
References
[1] A. W. Vyawahare, Near pseudo Smarandache funchion, Smarandache Notion Journal,
14(2004), 42-59.
[2] Yongfeng Zhang, On a near pseudo Smarandache funchion, Scientia Magna, 1(2007),
No.1, 98-101.
[3] Hai Yang and Ruiqin Fu, On the mean value of the Near Pseudo Smarandanche Func-
tion, Scientia Magna, 2(2006), No.4, 35-39.