Divisors of a $ k $-multiperfect number
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Let $ n $ be a $ k $-multiperfect number. Denote by $ d_m $ its $ m $ smallest divisor, and $ n_{m} $ the number of divisors of $ n $ divisible by $ d_m $. Is there for all $ 2leq mleqtau(n) $ an integer $ s(m) $ such that $ n_m=d_{s(m)} $? Does the function $ s $ admit a fixed point ?
Edit : actually $ n_{m}=tau(n/d_{m}) $. A sufficient condition for a number $ n $ to fulfill $ d=tau(n/d) $ is to take $ n=prod_{r}p_{r}^{p_{sigma(r)}-1} $ with $ sigma $ a permutation of the set of primes $ p $ such that $v_{p}(n)geq 1 $ and $d=prod_{p, v_{p}(n)geq 1}p $ .
number-theory perfect-numbers
asked Dec 23 '18 at 21:02
Sylvain JulienSylvain Julien
1,135918
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add a comment |
$begingroup$
Let $ n $ be a $ k $-multiperfect number. Denote by $ d_m $ its $ m $ smallest divisor, and $ n_{m} $ the number of divisors of $ n $ divisible by $ d_m $. Is there for all $ 2leq mleqtau(n) $ an integer $ s(m) $ such that $ n_m=d_{s(m)} $? Does the function $ s $ admit a fixed point ?
Edit : actually $ n_{m}=tau(n/d_{m}) $. A sufficient condition for a number $ n $ to fulfill $ d=tau(n/d) $ is to take $ n=prod_{r}p_{r}^{p_{sigma(r)}-1} $ with $ sigma $ a permutation of the set of primes $ p $ such that $v_{p}(n)geq 1 $ and $d=prod_{p, v_{p}(n)geq 1}p $ .
number-theory perfect-numbers
asked Dec 23 '18 at 21:02
Sylvain JulienSylvain Julien
1,135918
$endgroup$
$begingroup$
I am not sure, but maybe you could use a result in Dagal's A Lower Bound for $tau(n)$ of $k$-Multiperfect Number?
$endgroup$
– Jose Arnaldo Bebita Dris
Feb 4 at 4:53
add a comment |
$begingroup$
Let $ n $ be a $ k $-multiperfect number. Denote by $ d_m $ its $ m $ smallest divisor, and $ n_{m} $ the number of divisors of $ n $ divisible by $ d_m $. Is there for all $ 2leq mleqtau(n) $ an integer $ s(m) $ such that $ n_m=d_{s(m)} $? Does the function $ s $ admit a fixed point ?
Edit : actually $ n_{m}=tau(n/d_{m}) $. A sufficient condition for a number $ n $ to fulfill $ d=tau(n/d) $ is to take $ n=prod_{r}p_{r}^{p_{sigma(r)}-1} $ with $ sigma $ a permutation of the set of primes $ p $ such that $v_{p}(n)geq 1 $ and $d=prod_{p, v_{p}(n)geq 1}p $ .
number-theory perfect-numbers
asked Dec 23 '18 at 21:02
Sylvain JulienSylvain Julien
1,135918
$endgroup$
Let $ n $ be a $ k $-multiperfect number. Denote by $ d_m $ its $ m $ smallest divisor, and $ n_{m} $ the number of divisors of $ n $ divisible by $ d_m $. Is there for all $ 2leq mleqtau(n) $ an integer $ s(m) $ such that $ n_m=d_{s(m)} $? Does the function $ s $ admit a fixed point ?
Edit : actually $ n_{m}=tau(n/d_{m}) $. A sufficient condition for a number $ n $ to fulfill $ d=tau(n/d) $ is to take $ n=prod_{r}p_{r}^{p_{sigma(r)}-1} $ with $ sigma $ a permutation of the set of primes $ p $ such that $v_{p}(n)geq 1 $ and $d=prod_{p, v_{p}(n)geq 1}p $ .
number-theory perfect-numbers
number-theory perfect-numbers
asked Dec 23 '18 at 21:02
Sylvain JulienSylvain Julien
1,135918
asked Dec 23 '18 at 21:02
Sylvain JulienSylvain Julien
1,135918
edited Dec 23 '18 at 22:19
Sylvain Julien
asked Dec 23 '18 at 21:02
Sylvain JulienSylvain Julien
1,135918
asked Dec 23 '18 at 21:02
Sylvain JulienSylvain Julien
1,135918
asked Dec 23 '18 at 21:02
Sylvain JulienSylvain Julien
1,135918
1,135918
$begingroup$
I am not sure, but maybe you could use a result in Dagal's A Lower Bound for $tau(n)$ of $k$-Multiperfect Number?
$endgroup$
– Jose Arnaldo Bebita Dris
Feb 4 at 4:53
add a comment |
$begingroup$
I am not sure, but maybe you could use a result in Dagal's A Lower Bound for $tau(n)$ of $k$-Multiperfect Number?
$endgroup$
– Jose Arnaldo Bebita Dris
Feb 4 at 4:53
$begingroup$
I am not sure, but maybe you could use a result in Dagal's A Lower Bound for $tau(n)$ of $k$-Multiperfect Number?
$endgroup$
– Jose Arnaldo Bebita Dris
Feb 4 at 4:53
$begingroup$
I am not sure, but maybe you could use a result in Dagal's A Lower Bound for $tau(n)$ of $k$-Multiperfect Number?
$endgroup$
– Jose Arnaldo Bebita Dris
Feb 4 at 4:53
add a comment |
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$begingroup$
I am not sure, but maybe you could use a result in Dagal's A Lower Bound for $tau(n)$ of $k$-Multiperfect Number?
$endgroup$
– Jose Arnaldo Bebita Dris
Feb 4 at 4:53