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Note on the h th power means of distinct prime divisors of composite positive integers
Suppose that h is a positive integer. For an integer n ≥ 2 , define P h ( n ) = ( 1 ω ( n ) ∑ p ∣ n p prime p h ) 1 / h , where ω ( n ) denotes the number of distinct prime divisors of n . Let A h ( x ) be the set of all positive integers n ≤ x with ω ( n ) > 1 such that P h ( n ) is prime and P...
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| Published in: | Applied mathematics letters 2011-09, Vol.24 (9), p.1486-1490 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites |
| Online Access: | Get full text |
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| Summary: | Suppose that
h
is a positive integer. For an integer
n
≥
2
, define
P
h
(
n
)
=
(
1
ω
(
n
)
∑
p
∣
n
p
prime
p
h
)
1
/
h
,
where
ω
(
n
)
denotes the number of distinct prime divisors of
n
. Let
A
h
(
x
)
be the set of all positive integers
n
≤
x
with
ω
(
n
)
>
1
such that
P
h
(
n
)
is prime and
P
h
(
n
)
∣
n
. In this paper, we prove that
x
exp
(
2
h
log
x
log
log
x
)
≤
|
A
h
(
x
)
|
≤
x
exp
(
(
1
/
2
)
log
x
log
log
x
)
,
which generalizes a result of Luca and Pappalardi for
h
=
1
. |
|---|---|
| ISSN: | 0893-9659 1873-5452 |
| DOI: | 10.1016/j.aml.2011.03.015 |