Journal of Inequalities in Pure and
Applied Mathematics
A NOTE ON SUMS OF POWERS WHICH HAVE A FIXED
NUMBER OF PRIME FACTORS
volume 6, issue 2, article 31,
2005.
RAFAEL JAKIMCZUK
Department of Mathematics
Universidad Nacional de Luján
Argentina
Received 05 October, 2004;
accepted 16 February, 2005.
Communicated by: L. Tóth
EMail: jakimczu@mail.unlu.edu.ar
Abstract
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2000
Victoria University
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Abstract
Let us denote by cn,k the sequence of numbers which have in its factorization
kPprime factorsP(k ≥ 1), we obtain in short proofs asymptotic formulas for cn,k ,
n
α
α
i=1 ci,k and ci,k ≤ x ci,k . We generalize the work by T. Sálat y S. Znam when
k = 1 (see reference [2]).
2000 Mathematics Subject Classification: 11N25, 11N37.
Key words: Sums of powers, Numbers with k prime factors.
A Note On Sums Of Powers
Which Have A Fixed Number Of
Prime Factors
Rafael Jakimczuk
Let πk (x) be the number of these numbers not exceeding x, it was proved by
Landau [1] that
(1)
lim
πk (x)
k−1
x→∞ x(log log x)
(k−1)! log x
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=1
Note that if k = 1 then π1 (x) = π(x), cn,1 = pn , and equation (1) is the prime
number theorem.
Theorem 1. The following asymptotic formula holds:
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(2)
cn,k ∼
(k − 1)!n log n
(log log n)k−1
.
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J. Ineq. Pure and Appl. Math. 6(2) Art. 31, 2005
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Proof. If k = 1 the formula is true, since in this case (2) is the prime number
theorem pn ∼ n log n. Suppose k ≥ 2. If we put x = cn,k and substitute into
(1) we find that
(3)
lim
(k − 1)!n log cn,k
n→∞
cn,k (log log cn,k )k−1
= 1.
Writing
cn,k =
(4)
(k − 1)!n log n
k−1
(log log n)
A Note On Sums Of Powers
Which Have A Fixed Number Of
Prime Factors
f (n)
and substituting (4) into (3) we obtain
(5)
lim
n→∞
log cn,k (log log n)k−1
log n f (n) (log log cn,k )k−1
Rafael Jakimczuk
= 1.
From equation (1) we find that
(6)
πk (x)
lim
= ∞.
x→∞ π(x)
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Assume that the inequalities cn,k ≥ pn have infinitely many solutions, then we
have π(cn,k ) ≥ π(pn ) = n = πk (cn,k ), which contradicts (6). Hence for all
sufficiently large n we have cn,k < pn . On the other hand, clearly n ≤ cn,k .
Therefore n ≤ cn,k ≤ pn , that is log n ≤ log cn,k ≤ log pn , and we find that
(7)
1≤
log pn
log cn,k
≤
.
log n
log n
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J. Ineq. Pure and Appl. Math. 6(2) Art. 31, 2005
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From (7) and the prime number theorem pn ∼ n log n, we obtain
lim
(8)
n→∞
log cn,k
= 1.
log n
From (5) and (8) we find that
lim f (n) = 1.
(9)
n→∞
To finish, (9) and (4) give (2). The theorem is thus proved.
The following proposition is well known, we use it as a lemma
P
P
Lemma 2. Let ∞
ai and ∞
i=1P
i=1 bi be two series of positive terms such that
∞
an
lim bn = 1. Then if i=1 bi is divergent, the following limit holds
A Note On Sums Of Powers
Which Have A Fixed Number Of
Prime Factors
Rafael Jakimczuk
n→∞
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Pn
ai
lim Pi=1
= 1.
n
n→∞
i=1 bi
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Theorem 3. Let k ≥ 1 and let α be a positive number. The following asymptotic
formula holds
n
X
(10)
cαi,k ∼
i=1
i=1
cαi,k
and
.
(α + 1) (log log n)α(k−1)
1+2+
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((k − 1)!)α nα+1 logα n
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Proof. Let us consider the following two series:
∞
X
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∞
X
(k − 1)! i log i
i=3
(log log i)k−1
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!α
.
J. Ineq. Pure and Appl. Math. 6(2) Art. 31, 2005
http://jipam.vu.edu.au
Since the function
that
(11)
1+2+
(k−1)! t log t
(log log t)k−1
α
is increasing from a certain value of t, we find
n
X
(k − 1)! i log i
i=3
(log log i)k−1
Z
=
3
n
!α
(k − 1)! t log t
!α
dt + O
(log log t)k−1
n log n
!α !
(log log n)k−1
.
On the other hand, from the L’Hospital rule
!α
Z n
(k − 1)! t log t
((k − 1)!)α nα+1 logα n
(12)
dt
∼
.
(log log t)k−1
(α + 1) (log log n)α (k−1)
3
A Note On Sums Of Powers
Which Have A Fixed Number Of
Prime Factors
Rafael Jakimczuk
Title Page
Equation (10) is an immediate consequence of (11), (12) and the lemma.
The theorem is thus proved.
Theorem 4. Let k ≥ 1 and let α be a positive number. The following asymptotic
formula holds
(13)
X
cαi,k ∼
ci,k ≤ x
xα+1 (log log x)k−1
.
(α + 1) (k − 1)! log x
lim
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Proof. Equation (3) can be written in the form
(14)
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n
k−1
n→∞ cn,k (log log cn,k )
(k−1)! log cn,k
= 1.
J. Ineq. Pure and Appl. Math. 6(2) Art. 31, 2005
http://jipam.vu.edu.au
From (8) we obtain
log log cn,k
= 1.
n→∞ log log n
lim
(15)
Substituting (14), (8) and (15) into (10) we find that
X
(16)
cαi,k ∼
ci,k ≤ cn,k
k−1
cα+1
n,k (log log cn,k )
.
(α + 1) (k − 1)! log cn,k
A Note On Sums Of Powers
Which Have A Fixed Number Of
Prime Factors
Equation (2) gives cn,k ∼ cn+1,k , therefore
X
(17)
cαi,k ∼
ci,k ≤ cn,k
∼
k−1
cα+1
n+1,k (log log cn+1,k )
(α + 1) (k − 1)! log cn+1,k
Rafael Jakimczuk
k−1
cα+1
n,k (log log cn,k )
.
(α + 1) (k − 1)! log cn,k
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Since the function
xα+1 (log log x)k−1
(α + 1) (k − 1)! log x
is increasing from a certain value of x, we have for all n sufficiently large
P
P α
P
cαi,k
ci,k
cαi,k
(18)
ci,k ≤ cn,k
k−1
cα+1
n,k (log log cn,k )
(α+1) (k−1)! log cn,k
≤
ci,k ≤ x
xα+1 (log log x)k−1
(α+1) (k−1)! log x
6
ci,k ≤ cn,k
k−1
cα+1
n+1,k (log log cn+1,k )
(α+1) (k−1)! log cn+1,k
where cn,k ≤ x < cn+1,k .
To finish, (17) and (18) give (13). The theorem is proved.
,
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J. Ineq. Pure and Appl. Math. 6(2) Art. 31, 2005
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Note. The case k = 1 was studied in the reference [2]. In this case (9) and (13)
become
n
X
X
xα+1
nα+1 logα n
,
pαi ∼
.
pαi ∼
(α + 1)
(α + 1) log x
i=1
p ≤x
i
Using equation (2) and the lemma, we can prove (as above) other theorems, for
example the following:
Theorem 5. The following asymptotic formulas holds
∞
X
(log log n)k
1
∼
c
k!
n=1 n,k
and
When k = 1, this theorem is well known.
X
cn,k ≤ x
1
cn,k
k
∼
(log log x)
.
k!
A Note On Sums Of Powers
Which Have A Fixed Number Of
Prime Factors
Rafael Jakimczuk
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References
[1] G.H. HARDY AND E.M. WRIGHT, An Introduction to Number Theory, 4th
Ed.1960. Chapter XXII.
[2] T. SÁLAT AND S. ZNAM, On the sums of prime powers, Acta Fac. Rer.
Nat. Univ. Com. Math., 21 (1968), 21–25.
A Note On Sums Of Powers
Which Have A Fixed Number Of
Prime Factors
Rafael Jakimczuk
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J. Ineq. Pure and Appl. Math. 6(2) Art. 31, 2005
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