How many integers whose total number of prime factors is prime are there below x?
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Let $ Omega(n) $ be the total number of prime factors of a positive integer $ n $. Denote by $mathbb{P}_{Omega}(x) $ the number of positive integers $ n $ not exceeding $ x $ such that $ Omega(n)inmathbb{P} $. Is an upper bound for this function known ? If yes, is the fact that all non trivial partitions of a prime contain at least $ 2 $ distinct summands anyhow used in establishing this upper bound ?
number-theory prime-numbers
asked Dec 31 '18 at 20:00
Sylvain JulienSylvain Julien
1,150918
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show 1 more comment
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Let $ Omega(n) $ be the total number of prime factors of a positive integer $ n $. Denote by $mathbb{P}_{Omega}(x) $ the number of positive integers $ n $ not exceeding $ x $ such that $ Omega(n)inmathbb{P} $. Is an upper bound for this function known ? If yes, is the fact that all non trivial partitions of a prime contain at least $ 2 $ distinct summands anyhow used in establishing this upper bound ?
number-theory prime-numbers
asked Dec 31 '18 at 20:00
Sylvain JulienSylvain Julien
1,150918
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$begingroup$
$P_{Omega}(x,0)$ in here mathoverflow.net/questions/297785/…
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– Collag3n
Dec 31 '18 at 21:04
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Thank but I think in that link $ k $ is a number while I meant $ mathbb{A}_{f}(x) $ with $ f $ a function to denote the number of positive integers $ n $ below $ x $ such that the numbers $ f(n) $ belong to the set $ mathbb{A} $ .
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– Sylvain Julien
Dec 31 '18 at 21:43
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That way, $ pi(x) $ is $ mathbb{P}_{Id}(x) $ .
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– Sylvain Julien
Dec 31 '18 at 21:48
1
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Ok, so $P_{Omega}(x) = P_2(x,0) + P_3(x,0) + P_5(x,0) + P_7(x,0)....$ up to max $P_{lfloor log_{p_{(a+1)}}(x) rfloor}(x,0)$ ?
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– Collag3n
Dec 31 '18 at 21:59
1
$begingroup$
up to $x=15$ this is exactly $x-pi(x)-1$
$endgroup$
– Collag3n
Dec 31 '18 at 23:11
|
show 1 more comment
$begingroup$
Let $ Omega(n) $ be the total number of prime factors of a positive integer $ n $. Denote by $mathbb{P}_{Omega}(x) $ the number of positive integers $ n $ not exceeding $ x $ such that $ Omega(n)inmathbb{P} $. Is an upper bound for this function known ? If yes, is the fact that all non trivial partitions of a prime contain at least $ 2 $ distinct summands anyhow used in establishing this upper bound ?
number-theory prime-numbers
asked Dec 31 '18 at 20:00
Sylvain JulienSylvain Julien
1,150918
$endgroup$
Let $ Omega(n) $ be the total number of prime factors of a positive integer $ n $. Denote by $mathbb{P}_{Omega}(x) $ the number of positive integers $ n $ not exceeding $ x $ such that $ Omega(n)inmathbb{P} $. Is an upper bound for this function known ? If yes, is the fact that all non trivial partitions of a prime contain at least $ 2 $ distinct summands anyhow used in establishing this upper bound ?
number-theory prime-numbers
number-theory prime-numbers
asked Dec 31 '18 at 20:00
Sylvain JulienSylvain Julien
1,150918
asked Dec 31 '18 at 20:00
Sylvain JulienSylvain Julien
1,150918
asked Dec 31 '18 at 20:00
Sylvain JulienSylvain Julien
1,150918
asked Dec 31 '18 at 20:00
Sylvain JulienSylvain Julien
1,150918
asked Dec 31 '18 at 20:00
Sylvain JulienSylvain Julien
1,150918
1,150918
$begingroup$
$P_{Omega}(x,0)$ in here mathoverflow.net/questions/297785/…
$endgroup$
– Collag3n
Dec 31 '18 at 21:04
$begingroup$
Thank but I think in that link $ k $ is a number while I meant $ mathbb{A}_{f}(x) $ with $ f $ a function to denote the number of positive integers $ n $ below $ x $ such that the numbers $ f(n) $ belong to the set $ mathbb{A} $ .
$endgroup$
– Sylvain Julien
Dec 31 '18 at 21:43
$begingroup$
That way, $ pi(x) $ is $ mathbb{P}_{Id}(x) $ .
$endgroup$
– Sylvain Julien
Dec 31 '18 at 21:48
1
$begingroup$
Ok, so $P_{Omega}(x) = P_2(x,0) + P_3(x,0) + P_5(x,0) + P_7(x,0)....$ up to max $P_{lfloor log_{p_{(a+1)}}(x) rfloor}(x,0)$ ?
$endgroup$
– Collag3n
Dec 31 '18 at 21:59
1
$begingroup$
up to $x=15$ this is exactly $x-pi(x)-1$
$endgroup$
– Collag3n
Dec 31 '18 at 23:11
|
show 1 more comment
$begingroup$
$P_{Omega}(x,0)$ in here mathoverflow.net/questions/297785/…
$endgroup$
– Collag3n
Dec 31 '18 at 21:04
$begingroup$
Thank but I think in that link $ k $ is a number while I meant $ mathbb{A}_{f}(x) $ with $ f $ a function to denote the number of positive integers $ n $ below $ x $ such that the numbers $ f(n) $ belong to the set $ mathbb{A} $ .
$endgroup$
– Sylvain Julien
Dec 31 '18 at 21:43
$begingroup$
That way, $ pi(x) $ is $ mathbb{P}_{Id}(x) $ .
$endgroup$
– Sylvain Julien
Dec 31 '18 at 21:48
1
$begingroup$
Ok, so $P_{Omega}(x) = P_2(x,0) + P_3(x,0) + P_5(x,0) + P_7(x,0)....$ up to max $P_{lfloor log_{p_{(a+1)}}(x) rfloor}(x,0)$ ?
$endgroup$
– Collag3n
Dec 31 '18 at 21:59
1
$begingroup$
up to $x=15$ this is exactly $x-pi(x)-1$
$endgroup$
– Collag3n
Dec 31 '18 at 23:11
$begingroup$
$P_{Omega}(x,0)$ in here mathoverflow.net/questions/297785/…
$endgroup$
– Collag3n
Dec 31 '18 at 21:04
$begingroup$
$P_{Omega}(x,0)$ in here mathoverflow.net/questions/297785/…
$endgroup$
– Collag3n
Dec 31 '18 at 21:04
$begingroup$
Thank but I think in that link $ k $ is a number while I meant $ mathbb{A}_{f}(x) $ with $ f $ a function to denote the number of positive integers $ n $ below $ x $ such that the numbers $ f(n) $ belong to the set $ mathbb{A} $ .
$endgroup$
– Sylvain Julien
Dec 31 '18 at 21:43
$begingroup$
Thank but I think in that link $ k $ is a number while I meant $ mathbb{A}_{f}(x) $ with $ f $ a function to denote the number of positive integers $ n $ below $ x $ such that the numbers $ f(n) $ belong to the set $ mathbb{A} $ .
$endgroup$
– Sylvain Julien
Dec 31 '18 at 21:43
$begingroup$
That way, $ pi(x) $ is $ mathbb{P}_{Id}(x) $ .
$endgroup$
– Sylvain Julien
Dec 31 '18 at 21:48
$begingroup$
That way, $ pi(x) $ is $ mathbb{P}_{Id}(x) $ .
$endgroup$
– Sylvain Julien
Dec 31 '18 at 21:48
1
1
$begingroup$
Ok, so $P_{Omega}(x) = P_2(x,0) + P_3(x,0) + P_5(x,0) + P_7(x,0)....$ up to max $P_{lfloor log_{p_{(a+1)}}(x) rfloor}(x,0)$ ?
$endgroup$
– Collag3n
Dec 31 '18 at 21:59
$begingroup$
Ok, so $P_{Omega}(x) = P_2(x,0) + P_3(x,0) + P_5(x,0) + P_7(x,0)....$ up to max $P_{lfloor log_{p_{(a+1)}}(x) rfloor}(x,0)$ ?
$endgroup$
– Collag3n
Dec 31 '18 at 21:59
1
1
$begingroup$
up to $x=15$ this is exactly $x-pi(x)-1$
$endgroup$
– Collag3n
Dec 31 '18 at 23:11
$begingroup$
up to $x=15$ this is exactly $x-pi(x)-1$
$endgroup$
– Collag3n
Dec 31 '18 at 23:11
|
show 1 more comment
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$begingroup$
$P_{Omega}(x,0)$ in here mathoverflow.net/questions/297785/…
$endgroup$
– Collag3n
Dec 31 '18 at 21:04
$begingroup$
Thank but I think in that link $ k $ is a number while I meant $ mathbb{A}_{f}(x) $ with $ f $ a function to denote the number of positive integers $ n $ below $ x $ such that the numbers $ f(n) $ belong to the set $ mathbb{A} $ .
$endgroup$
– Sylvain Julien
Dec 31 '18 at 21:43
$begingroup$
That way, $ pi(x) $ is $ mathbb{P}_{Id}(x) $ .
$endgroup$
– Sylvain Julien
Dec 31 '18 at 21:48
1
$begingroup$
Ok, so $P_{Omega}(x) = P_2(x,0) + P_3(x,0) + P_5(x,0) + P_7(x,0)....$ up to max $P_{lfloor log_{p_{(a+1)}}(x) rfloor}(x,0)$ ?
$endgroup$
– Collag3n
Dec 31 '18 at 21:59
1
$begingroup$
up to $x=15$ this is exactly $x-pi(x)-1$
$endgroup$
– Collag3n
Dec 31 '18 at 23:11