PROPERTIES OF NON POWERFUL NUMBERS
VLAD COPIL
LAURENŢIU PANAITOPOL
University "Spiru Haret"
Department of Mathematics
Str. Ion Ghica, 13, 030045
Bucharest, Romania
University of Bucharest
Department of Mathematics
Str. Academiei 14, 010014
Bucharest, Romania
EMail: vladcopil@gmail.com
EMail: pan@al.math.unibuc.ro
Properties of Non Powerful
Numbers
Vlad Copil and
Laurenţiu Panaitopol
vol. 9, iss. 1, art. 29, 2008
Title Page
Received:
11 January, 2008
Accepted:
10 March, 2008
Communicated by:
L. Tóth
2000 AMS Sub. Class.:
11B83, 11P32, 26D07.
Key words:
Powerful numbers, Sequences, Inequalities, Goldbach’s conjecture.
Abstract:
In this paper we study some properties of non powerful numbers. We evaluate
the n-th non powerful number and prove for the sequence of non powerful numbers some theorems that are related to the sequence of primes: Landau, Mandl,
Scherk. Related to the conjecture of Goldbach, we prove that every positive integer ≥ 3 is the sum between a prime and a non powerful number.
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Contents
1
Introduction
3
2
Inequalities for vn
5
3
Some Properties of the Sequence of Non Powerful Numbers
11
Properties of Non Powerful
Numbers
Vlad Copil and
Laurenţiu Panaitopol
vol. 9, iss. 1, art. 29, 2008
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1.
Introduction
A positive integer v is called non powerful if there exists a prime p such that p|v and
p2 - v.
Otherwise, if v has the canonical decomposition v = q1α1 · · · · · qrαr , there exists
j ∈ {1, 2, . . . , r} such that αj = 1.
It results that v can be written uniquely as v = f · u, where f is squarefree, u is
powerful and (f, u) = 1.
In this paper we use the following notations:
• K(x)= the number of powerful numbers less than or equal to x
• C(x)= the number of non powerful numbers less than or equal to x
Properties of Non Powerful
Numbers
Vlad Copil and
Laurenţiu Panaitopol
vol. 9, iss. 1, art. 29, 2008
Title Page
• vn is the n-th non powerful number
Contents
We use a special case of a classical formula:
Theorem A. If h ∈ C 1 , g is continuous, a is powerful and
X
G(x) =
g(v),
a≤v≤x
v non powerful
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then
X
a≤v≤x
v non powerful
Z
h(v)g(v) = h(x)G(x) −
x
h0 (t)G(t)dt.
a
G. Mincu and L. Panaitopol proved [5] the following.
Close
Theorem B.
and
√
√
K(x) ≥ c x − 1.83522 3 x
√
√
K(x) ≤ c x − 1.207684 3 x
for x ≥ 961
for x ≥ 4.
As C(x) = [x] − K(x) it results that
√
√
√
√
(1.1)
[x] − c x + 1.207684 3 x ≤ C(x) ≤ [x] − c x + 1.83522 3 x
the first inequality being true for x ≥ 4, while the second one is true for x ≥ 961.
We also use
Theorem C. We have the relation
Properties of Non Powerful
Numbers
Vlad Copil and
Laurenţiu Panaitopol
vol. 9, iss. 1, art. 29, 2008
Title Page
1
3
1
ζ(3/2) √
ζ(2/3) √
3
x + O x 6 exp(−c1 log 5 (log log x)− 5 .
x+
K(x) =
ζ(2)
ζ(3)
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Inequalities for vn
2.
Theorem 2.1. We have the relation
√
√
vn > n + c n − a 3 n
for n ≥ 88,
where a = 1.83522.
Proof. If we put x = vn in the second inequality from (1.1), it results that
√
√
n ≤ vn − c vn + a 3 vn
for n ≥ 4.
√
√
√
√
Let f (x) = x − c x + a 3 x − n and x0n = n + c n − k 3 n. As f (vn ) > 0, and
f is increasing, if we prove that f (x0n ) < 0, it results that vn > x0n .
Denote g(n) = f (x0n ). Proving that f (x0n ) < 0 is equivalent with proving that
g(n) < 0. Therefore we intend to prove that
q
q
√
√
√
√
√
√
3
g(n) = c n − k 3 n − c n + c n − k 3 n + a n + c n − k 3 n < 0.
We use the following relations for x > 0:
(2.1)
x x2
x √
1+ > 1+x>1+ −
2
2
8
Laurenţiu Panaitopol
vol. 9, iss. 1, art. 29, 2008
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and
x √
x x2
> 31+x>1+ − .
3
3
9
0
Putting x = xn in (2.1) gives
2
√ q
√
√
√
√
c
k
c
k
n
c
k
3
√ −√
n+ − √
> n+c n−k n > n+ − √
−
,
3
2 26n
8
2 26n
n
n2
(2.2)
Properties of Non Powerful
Numbers
Vlad Copil and
1+
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while x = x0n gives from (2.2)
2
√
q
3
√
√
√
√
k
c
k
c
k
n
c
3
3
3
3
√ −√
.
− √ > n + c n − k n > n+ √
− √ −
n+ √
3
n
36n 33n
36n 33n 9
n2
Using the previous relations in the expression of g(n) yields
2
√ √
√
√ c2 ck c n
√
k
ac
ak
c
3
√ −√
g(n) < c n−k n−c n− + √
+
+a 3 n+ √
− √
.
3
6
6
2 2 n
8
n
3 n 33n
n2
In order to prove g(n) > 0 it is enough to prove that
2
√ √
c
k
ak
c n
c2
ck ac
1
3
√ −√
√
< 0.
− √ +
(a − k) n − +
+
3
6
8
2
2
3
n
n 33n
n2
The best result is obtained by taking k = a, therefore
2
√ c
a
a2
c n
c2 5 ac
√ −√
< 0.
− √ +
− + √
3
8
2
6 6n 33n
n
n2
As
√c
n
>
a
√
3 2
n
for n ≥ 1, it is enough to prove that
√
c2
a2
5ac
c n c2
√
√ .
·
<
+
+
8
n
2
66n
33n
The last relation is true because
3 6
c3
a2
3c
√ < √
⇔
<n
3
8a2
8 n
3 n
and
5ac
c2
√
<
⇔
2
66n
5a
3c
Laurenţiu Panaitopol
vol. 9, iss. 1, art. 29, 2008
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that holds for n ≥ 816
6
<n
Properties of Non Powerful
Numbers
Vlad Copil and
that holds for n ≥ 8.
In conclusion, we have
√
√
vn > n + c n − a 3 n
for n ≥ 816. Verifications done using the computer allow us to lower the bound to
n ≥ 88.
Theorem 2.2. We have the relation
√
√
vn < n + c n − 3 n
for n ≥ 1.
Proof. If we put x = vn in the first inequality from (1.1), it results that
√
√
n > vn − c v n + α 3 v n ,
where α = 1.207684.
√
√
√
√
Let f (x) = x − c x + α 3 x − n and x00n = n + c n − h 3 n. We have f (vn ) < 0,
f is increasing, so if we prove that f (x00n ) > 0, it results that vn < x00n .
Denote g(n) = f (x00n ). Proving that f (x00n ) > 0 is equivalent to proving that
g(n) > 0. Therefore we have to prove that
q
q
√
√
√
√
√
√
3
g(n) = n + c n − h 3 n − c n + c n − h 3 n + α n + c n − h 3 n − n < 0.
Using the relations (2.1) and (2.2) as we did in the proof of Theorem 2.1, gives
2
√
2
3
√
√
√
√
c
ch
αc
n
c
αh
α
h
√ −√
c n−h 3 n−c n− + √
+α 3 n+ √
− √ −
> 0.
3
2 26n
9
36n 33n
n
n2
The previous relation is equivalent to
2
√ √
1
c
ch αc
c2 α 3 n
h
αh
3
√
√
√
√ .
+
+
n(α − h) +
>
−
+
3
6
2
3
2
9
n
n
33n
n2
Properties of Non Powerful
Numbers
Vlad Copil and
Laurenţiu Panaitopol
vol. 9, iss. 1, art. 29, 2008
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Thus, it is enough to prove that for h < α
√
√
α 3 n c2
c
h α
c2
αh
3
· .
+
+
>
+ √
n(α − h) + √
6
9
n
3
2
33n
n 2
√
2
We have 3 n(α − h) > c2 , if
3
c2
.
(2.3)
n>
2(α − h)
Laurenţiu Panaitopol
vol. 9, iss. 1, art. 29, 2008
It remains to prove that
c
√
6
n
Properties of Non Powerful
Numbers
Vlad Copil and
h α
+
2
3
αh
αc2
√ .
> √
+
3 3 n 9 3 n2
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2
αc
α
√
√
Therefore it is enough to prove that c h2 > 3αh
6 n and that c 3 > 9 n ; both the relations
are true for n ≥ 1.
In conclusion, the condition (2.3) gives the lower bound for realizing the inequality from Theorem 2.2: we take h = 1 so n > 1471. Verification using the computer
allows us to take n ≥ 1.
Theorem 2.3. There exists c2 > 0 such that
3
ζ(3/2) √
ζ(2/3) √
− 15
3
5
vn = n +
n+
n + O exp(−c2 log n(log log n)
.
ζ(3)
ζ(2)
Proof. We have C(x) = [x] − K(x), and put x = vn . It results that n = vn − K(vn );
we use Theorem C to evaluate K, and obtain
1
√
√
n = vn − c vn − b 3 vn + O n 6 g(n) ,
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3
5
where c = ζ(3/2)/ζ(3), b = ζ(2/3)/ζ(2) and g(n) = exp −c2 (log n) (log log n)
with c2 > 0 and g(n) → ∞ as n → ∞.
So
1
√
√
(2.4)
−n + vn − c vn − b 3 vn = O n 6 g(n) .
From Theorem 2.1 and 2.2 we have
√
√
√
√
n + c n − 1.83522 3 n < vn < n + c n − 3 n,
Properties of Non Powerful
Numbers
Vlad Copil and
Laurenţiu Panaitopol
therefore
vol. 9, iss. 1, art. 29, 2008
√
√
3
vn = n + c n − xn n,
(2.5)
with (xn )n≥1 bounded.
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It is known that
√
and
√
3
1+x=1+
1+x=1+
x x2
−
+ ···
2
8
Contents
x x2
−
+ ··· .
3
9
√
vn =
=
√
√
r
n
c
xn
1+ √ − √
3
n
n2
vn =
√
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c
xn
√ −√
3
n
n2
!
2
+ ···
so
√
II
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c
1
xn
n 1+ √ − √
−
3
2 n 2 n2 8
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Page 9 of 18
Therefore
(2.6)
−1
5
2
√ c
n
xn
c
xn
√ −√
n+ − √
−
+ ··· .
3
2 26n
8
n
n2
,
Close
In a similar manner, we get
(2.7)
√
3
vn =
√
3
2
√
3
xn
n
c
xn
c
√ −√
+ ··· .
n+ √
− √ −
3
9
36n 33n
n
n2
From (2.4), (2.6) and (2.7) it results that
2
√ cxn
c2
c n
xn
c
√ −√
c n − xn n − c n − + √
+
3
2
8
26n
n
n2
√
2
1
√
c
xn
bc
bxn
b3n
3
6 g(n)
√
√
√
+
·
·
·
=
O
n
−
.
+
+
−b n− √
3
9
n
36n 33n
n2
√
√
3
Therefore
√
Laurenţiu Panaitopol
vol. 9, iss. 1, art. 29, 2008
Title Page
√
3
− n(xn + b) = O n g(n) ,
1
6
which yields
(2.8)
Properties of Non Powerful
Numbers
Vlad Copil and
xn = −b + O
g(n)
√
6
n
.
From (2.5) and (2.8) we obtain
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√
√
3
√ vn = n + c n + b n + O g(n) 3 n .
In conclusion, there exists c2 > 0 such that
√
√
3
1
3
5
5
vn = n + c n + b n + O exp(−c2 log n(log log n) ) .
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3.
Some Properties of the Sequence of Non Powerful Numbers
In relation to the prime number distribution function, E. Landau [4] proved in 1909
that
π(2x) < 2π(x) for x ≥ x0 .
Afterwards J.B. Rosser and L. Schoenfeld proved [6] that
π(2x) < 2π(x) for all x > 2.
Properties of Non Powerful
Numbers
Vlad Copil and
Laurenţiu Panaitopol
In relation to this problem we can state the following result.
vol. 9, iss. 1, art. 29, 2008
Theorem 3.1. We have the relation
(3.1)
C(2x) ≥ 2C(x)
for all integers x ≥ 7.
Proof. Using Theorem B we obtain:
√
√
√
√
[x] − c x + 1.207684 3 x ≤ C(x) ≤ [x] − c x + 1.83522 3 x,
for x ≥ 961.
In order to prove (3.1) it is therefore sufficient to show that
√
√
√
√
3
[2x] − c 2x + 1.207864 2x ≥ 2[x] − 2c x + 3.67044 3 x.
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As [2x] ≥ 2[x], it is sufficient to show that
√
√
√
c x(2 − 2) > 2.14885307 3 x,
√
which is true if 6 x ≥ 1.687939, more precisely for x ≥ 24. Verifications done using
the computer show that Theorem 3.1 is true for every integer 8 ≤ x ≤ 961, which
concludes our proof.
Close
Remark 1. From Theorem 3.1 it follows that vn+1 < 2vn for every n ≥ 1.
The Mandl inequality [2] states that, for n ≥ 9
1
p1 + p2 + . . . + pn < npn ,
2
where pn is the n-the prime.
Properties of Non Powerful
Numbers
Vlad Copil and
Related to this inequality, we prove that for non powerful numbers
Theorem 3.2. We have for n ≥ 7 that
Laurenţiu Panaitopol
vol. 9, iss. 1, art. 29, 2008
1
v1 + v2 + . . . + vn > nvn .
2
(3.2)
Proof. Let n > C(961) + 1 = 912. In order to evaluate the sum
n
P
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vi , we use
Contents
i=1
Theorem A with h(t) = t, g(t) = 1 and a = 961. It follows that G(x) = C(x) −
C(961) and then we obtain
Z vn
n
X
vi = vn (n − C(961)) −
(C(t) − C(961))dt.
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Then
i=1
II
961
i=C(961)+1
n
X
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C(961)
vi =
X
i=1
Z
vn
vi + nvn − vn C(961) −
C(t)dt + C(961)(vn − 961).
961
Using Theorem B, we get a better upper bound for k 0 (x), namely
√
√
k 0 (x) ≤ x − c x + 1.83522 3 x for x ≥ 961.
Close
Therefore, it is enough to prove that
C(961)
X
Z
vn
vi + nvn − 961C(961) −
961
i=1
√
nvn
3
t + c t + 1.83522 t dt >
.
2
√
Integrating and making some further numerical calculus (C(961) = 911,
911
P
vi =
i=1
445213) lead us to
√
n vn 2c √
3
vn
−
+
vn − · 1.83522 3 vn > −463153.9136.
2
2
3
4
So, in order to prove (3.2), it is enough to prove that
√
n vn 2c √
3
−
+
vn − · 1.83522 3 vn > 0.
2
2
3
4
This is equivalent with proving that
√
4c √
3
vn < n +
vn − · 1.83522 3 vn .
3
2
Taking into account Theorem 2.2 and the fact that for n > C(961) + 1 we have
n < vn < 2n, it is enough to prove that
√
√
√
√
4c √
3
3
n+c n− 3n<n+
n − · 1.83522 · 2 · 3 n,
3
2
which is true for n ≥ 1565.
Verifications done with the computer, lead us to state that the theorem is true for
every n ≥ 1, excepting the case n = 7.
Properties of Non Powerful
Numbers
Vlad Copil and
Laurenţiu Panaitopol
vol. 9, iss. 1, art. 29, 2008
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The well known conjecture of Goldbach states that every even number is the sum
of two odd primes. Related to this problem, Chen Jing-Run has shown [1] using
the Large Sieve, that all large enough even numbers are the sum of a prime and the
product of at most two primes.
We present a weaker result, that has the advantage that is easily obtained and the
proof is true for every integer n ≥ 3.
Theorem 3.3. Every integer n ≥ 3 is the sum between a prime and a non powerful
number.
Proof. Let n ≥ 3 and pi the largest prime that does not exceed n. Thus pi < n ≤
pi+1 and
(
π(n) − 1, if n is prime,
i=
π(n), otherwise
Then we consider the numbers n − p1 , n − p2 , . . . , n − pi . We prove that one of
these i numbers is non powerful.
Suppose that all these i numbers are powerful. It results that
√
c n − 2 ≥ k(n − 2) ≥ i ≥ π(n) − 1.
Taking into account that π(x) >
x
log x
√
c n−2≥
√
for x ≥ 59, we obtain
n
− 1 for n ≥ 59.
log n
√
For n ≥ 4 we have c n − 2 > 2 n − 1, therefore it is enough to prove that
√
n
2 n≥
.
log n
√
But for n ≥ 75 we have 2 log n < n.
Properties of Non Powerful
Numbers
Vlad Copil and
Laurenţiu Panaitopol
vol. 9, iss. 1, art. 29, 2008
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Therefore the supposition we made (that n − p1 , n − p2 , . . . , n − pi are all powerful) is certainly false for n ≥ 75 and it results that every integer greater than 75
is the sum between a prime and a non powerful number. Direct computation leads
us to state that every integer n ≥ 3 is the sum between a prime and a non powerful
number.
In 1830, H. F. Scherk found that
p2n = 1 ± p1 ± p2 ± . . . ± p2n−2 + p2n−1
Properties of Non Powerful
Numbers
Vlad Copil and
Laurenţiu Panaitopol
and
vol. 9, iss. 1, art. 29, 2008
p2n+1 = 1 ± p1 ± p2 ± · · · ± p2n−1 + 2p2n .
The proof of these relations was first given by S. Pillai in 1928. W. Sierpinski gave
a proof of Scherk’s formulae in 1952, [7].
In relation to Scherk’s formulae, we have the following.
Theorem 3.4. For n ≥ 6, we have
vn = ±εn ± v1 ± v2 ± . . . ± vn−2 + vn−1
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where εn is 0 or 1.
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Proof. Following the method Sierpinski used in [7], we make an induction proof of
this theorem.
If n = 6, we have v6 = 10 and
1 = −2 − 3 + 5 − 6 + 7,
2 = 1 − 2 − 3 + 5 − 6 + 7,
3 = −2 − 3 − 5 + 6 + 7,
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4 = 1 − 2 − 3 − 5 + 6 + 7,
5=
2 − 3 + 5 − 6 + 7,
6 = 1 + 2 − 3 + 5 − 6 + 7,
7=
2 − 3 − 5 + 6 + 7,
8 = 1 + 2 − 3 − 5 + 6 + 7,
9 = −2 + 3 − 5 + 6 + 7,
10 = 1 − 2 + 3 − 5 + 6 + 7.
Properties of Non Powerful
Numbers
Vlad Copil and
Laurenţiu Panaitopol
Therefore every natural number less than or equal to 10 can be expressed in the
desired form.
We suppose the theorem is true for n and prove it for n + 1.
Let k be a positive integer less than or equal to vn+1 . Then, because vi+1 < 2vi
for every natural number i, we have
k ≤ vn+1 < 2vn ,
so
−vn < k − vn < vn .
vol. 9, iss. 1, art. 29, 2008
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It follows that 0 ≤ ±(k − vn ) < vn ; we can apply the induction hypothesis and
write ±(k − vn ) = ±εn ± v1 ± v2 ± . . . ± vn−2 + vn−1 . It will immediately follow
that there exist a choice of the signs + and − such that
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k = ±εn ± v1 ± v2 ± . . . ± vn−1 + vn .
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As vn ≤ vn+1 , we get
vn = ±εn ± v1 ± v2 ± . . . ± vn−2 + vn−1 .
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References
[1] JING-RUN CHEN, On the representation of a large even number as the sum of
a prime and the product of at most two primes, Sci. Sinica, 16 (1973), 157–176.
[2] P. DUSART, Sharper bounds for ψ, θ, π, pk , Rapport de Recherche, 1998.
[3] S.W. GOLOMB, Powerful numbers, Amer. Math. Monthly, 77 (1970), 848–852.
[4] E. LANDAU, Handbuch, Band I. Leipzig, (1909)
[5] G. MINCU AND L. PANAITOPOL, More about powerful numbers, in press.
Properties of Non Powerful
Numbers
Vlad Copil and
Laurenţiu Panaitopol
vol. 9, iss. 1, art. 29, 2008
[6] J.B. ROSSER AND L. SCHOENFELD, Abstracts of scientific communications,
Intern. Congr. Math. Moscow, (1966).
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[7] W. SIERPINSKI, Elementary Theory of Numbers, Warszawa, P.W.N., (1964).
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