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For every $k>2$, does there exist at least one k-perfect number?
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For every $k>2$, does there exist at least one number $n$ for which $sigma(n)=k*n$? What kind of heuristics can be done to shed light on this problem?
number-theory
asked Jul 21 at 6:12
Tyler Litch
412
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show 2 more comments
up vote
1
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favorite
For every $k>2$, does there exist at least one number $n$ for which $sigma(n)=k*n$? What kind of heuristics can be done to shed light on this problem?
number-theory
asked Jul 21 at 6:12
Tyler Litch
412
what is $sigma(n)$?
– uniquesolution
Jul 21 at 6:26
the sum of divisors function
– Tyler Litch
Jul 21 at 6:27
1
From the least numbers $n$ up to $k=11$ on Wikipedia, there seems to be an approximately exponential growth in the number of digits, very roughly $log npropto2.5^k$. From Grönwall's theorem, $$ limsup_nrightarrowinftyfracsigma(n)n,log log n=e^gamma;, $$ we might expect $log nproptomathrm e^k$.
– joriki
Jul 21 at 6:38
2
Nobody knows whether there is, for every $k>2$, an integer $n$ for which $sigma(n)=kn$.
– Gerry Myerson
Jul 21 at 7:15
Do we expect that there is? Why or why not?
– Tyler Litch
Jul 21 at 7:35
 |Â
show 2 more comments
up vote
1
down vote
favorite
up vote
1
down vote
favorite
For every $k>2$, does there exist at least one number $n$ for which $sigma(n)=k*n$? What kind of heuristics can be done to shed light on this problem?
number-theory
asked Jul 21 at 6:12
Tyler Litch
412
For every $k>2$, does there exist at least one number $n$ for which $sigma(n)=k*n$? What kind of heuristics can be done to shed light on this problem?
number-theory
asked Jul 21 at 6:12
Tyler Litch
412
asked Jul 21 at 6:12
Tyler Litch
412
asked Jul 21 at 6:12
Tyler Litch
412
asked Jul 21 at 6:12
Tyler Litch
412
412
what is $sigma(n)$?
– uniquesolution
Jul 21 at 6:26
the sum of divisors function
– Tyler Litch
Jul 21 at 6:27
1
From the least numbers $n$ up to $k=11$ on Wikipedia, there seems to be an approximately exponential growth in the number of digits, very roughly $log npropto2.5^k$. From Grönwall's theorem, $$ limsup_nrightarrowinftyfracsigma(n)n,log log n=e^gamma;, $$ we might expect $log nproptomathrm e^k$.
– joriki
Jul 21 at 6:38
2
Nobody knows whether there is, for every $k>2$, an integer $n$ for which $sigma(n)=kn$.
– Gerry Myerson
Jul 21 at 7:15
Do we expect that there is? Why or why not?
– Tyler Litch
Jul 21 at 7:35
 |Â
show 2 more comments
what is $sigma(n)$?
– uniquesolution
Jul 21 at 6:26
the sum of divisors function
– Tyler Litch
Jul 21 at 6:27
1
From the least numbers $n$ up to $k=11$ on Wikipedia, there seems to be an approximately exponential growth in the number of digits, very roughly $log npropto2.5^k$. From Grönwall's theorem, $$ limsup_nrightarrowinftyfracsigma(n)n,log log n=e^gamma;, $$ we might expect $log nproptomathrm e^k$.
– joriki
Jul 21 at 6:38
2
Nobody knows whether there is, for every $k>2$, an integer $n$ for which $sigma(n)=kn$.
– Gerry Myerson
Jul 21 at 7:15
Do we expect that there is? Why or why not?
– Tyler Litch
Jul 21 at 7:35
what is $sigma(n)$?
– uniquesolution
Jul 21 at 6:26
what is $sigma(n)$?
– uniquesolution
Jul 21 at 6:26
the sum of divisors function
– Tyler Litch
Jul 21 at 6:27
the sum of divisors function
– Tyler Litch
Jul 21 at 6:27
1
1
From the least numbers $n$ up to $k=11$ on Wikipedia, there seems to be an approximately exponential growth in the number of digits, very roughly $log npropto2.5^k$. From Grönwall's theorem, $$ limsup_nrightarrowinftyfracsigma(n)n,log log n=e^gamma;, $$ we might expect $log nproptomathrm e^k$.
– joriki
Jul 21 at 6:38
From the least numbers $n$ up to $k=11$ on Wikipedia, there seems to be an approximately exponential growth in the number of digits, very roughly $log npropto2.5^k$. From Grönwall's theorem, $$ limsup_nrightarrowinftyfracsigma(n)n,log log n=e^gamma;, $$ we might expect $log nproptomathrm e^k$.
– joriki
Jul 21 at 6:38
2
2
Nobody knows whether there is, for every $k>2$, an integer $n$ for which $sigma(n)=kn$.
– Gerry Myerson
Jul 21 at 7:15
Nobody knows whether there is, for every $k>2$, an integer $n$ for which $sigma(n)=kn$.
– Gerry Myerson
Jul 21 at 7:15
Do we expect that there is? Why or why not?
– Tyler Litch
Jul 21 at 7:35
Do we expect that there is? Why or why not?
– Tyler Litch
Jul 21 at 7:35
 |Â
show 2 more comments
1 Answer
1
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oldest
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up vote
1
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See Wikipedia's k-perfect numbers:
In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.
For a given natural number k, a number n is called k-perfect (or k-fold perfect) if and only if the sum of all positive divisors of n (the divisor function, Ã(n)) is equal to kn; a number is thus perfect if and only if it is 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number
$$
small
beginarrayrrl \
k & textSmallest k-perfect number & \
hline \
2 & 6 & 2 × 3 \
3 & 120 & 2^3 × 3 × 5 \
4 & 30240 & 2^5 × 3^3 × 5 × 7 \
5 & 14182439040 & 2^7 × 3^4 × 5 × 7 × 11^2 × 17 × 19 \
6 & text(21 digits) , 154345556085770649600 & 2^15 × 3^5 × 5^2 × 7^2 × 11 × 13 × 17 × 19 × 31 × 43 × 257 \
7 & ... & \
endarray
$$
See the Wikipedia page linked above for k-perfect numbers up to 11, they are too lengthy to fit on Stack Exchange pages. It's reasonable to expect that there are more but they haven't been discovered yet.
Other references:
Oeis - Multiply-Perfect Numbers
MathWorld - Multiperfect Number
"Moxham (2000) found the largest known multiperfect number, approximately equal to 7.3×10^(1345), on Feb. 13, 2000."
The "The Multiply Perfect Numbers Page" has a gzipped file from 2014 with 5311 MPNs, as of 2018-01-07 no new numbers have been discovered.
answered Jul 21 at 7:59
Rob
346112
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
See Wikipedia's k-perfect numbers:
In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.
For a given natural number k, a number n is called k-perfect (or k-fold perfect) if and only if the sum of all positive divisors of n (the divisor function, Ã(n)) is equal to kn; a number is thus perfect if and only if it is 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number
$$
small
beginarrayrrl \
k & textSmallest k-perfect number & \
hline \
2 & 6 & 2 × 3 \
3 & 120 & 2^3 × 3 × 5 \
4 & 30240 & 2^5 × 3^3 × 5 × 7 \
5 & 14182439040 & 2^7 × 3^4 × 5 × 7 × 11^2 × 17 × 19 \
6 & text(21 digits) , 154345556085770649600 & 2^15 × 3^5 × 5^2 × 7^2 × 11 × 13 × 17 × 19 × 31 × 43 × 257 \
7 & ... & \
endarray
$$
See the Wikipedia page linked above for k-perfect numbers up to 11, they are too lengthy to fit on Stack Exchange pages. It's reasonable to expect that there are more but they haven't been discovered yet.
Other references:
Oeis - Multiply-Perfect Numbers
MathWorld - Multiperfect Number
"Moxham (2000) found the largest known multiperfect number, approximately equal to 7.3×10^(1345), on Feb. 13, 2000."
The "The Multiply Perfect Numbers Page" has a gzipped file from 2014 with 5311 MPNs, as of 2018-01-07 no new numbers have been discovered.
answered Jul 21 at 7:59
Rob
346112
add a comment |Â
up vote
1
down vote
See Wikipedia's k-perfect numbers:
In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.
For a given natural number k, a number n is called k-perfect (or k-fold perfect) if and only if the sum of all positive divisors of n (the divisor function, Ã(n)) is equal to kn; a number is thus perfect if and only if it is 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number
$$
small
beginarrayrrl \
k & textSmallest k-perfect number & \
hline \
2 & 6 & 2 × 3 \
3 & 120 & 2^3 × 3 × 5 \
4 & 30240 & 2^5 × 3^3 × 5 × 7 \
5 & 14182439040 & 2^7 × 3^4 × 5 × 7 × 11^2 × 17 × 19 \
6 & text(21 digits) , 154345556085770649600 & 2^15 × 3^5 × 5^2 × 7^2 × 11 × 13 × 17 × 19 × 31 × 43 × 257 \
7 & ... & \
endarray
$$
See the Wikipedia page linked above for k-perfect numbers up to 11, they are too lengthy to fit on Stack Exchange pages. It's reasonable to expect that there are more but they haven't been discovered yet.
Other references:
Oeis - Multiply-Perfect Numbers
MathWorld - Multiperfect Number
"Moxham (2000) found the largest known multiperfect number, approximately equal to 7.3×10^(1345), on Feb. 13, 2000."
The "The Multiply Perfect Numbers Page" has a gzipped file from 2014 with 5311 MPNs, as of 2018-01-07 no new numbers have been discovered.
answered Jul 21 at 7:59
Rob
346112
add a comment |Â
up vote
1
down vote
up vote
1
down vote
See Wikipedia's k-perfect numbers:
In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.
For a given natural number k, a number n is called k-perfect (or k-fold perfect) if and only if the sum of all positive divisors of n (the divisor function, Ã(n)) is equal to kn; a number is thus perfect if and only if it is 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number
$$
small
beginarrayrrl \
k & textSmallest k-perfect number & \
hline \
2 & 6 & 2 × 3 \
3 & 120 & 2^3 × 3 × 5 \
4 & 30240 & 2^5 × 3^3 × 5 × 7 \
5 & 14182439040 & 2^7 × 3^4 × 5 × 7 × 11^2 × 17 × 19 \
6 & text(21 digits) , 154345556085770649600 & 2^15 × 3^5 × 5^2 × 7^2 × 11 × 13 × 17 × 19 × 31 × 43 × 257 \
7 & ... & \
endarray
$$
See the Wikipedia page linked above for k-perfect numbers up to 11, they are too lengthy to fit on Stack Exchange pages. It's reasonable to expect that there are more but they haven't been discovered yet.
Other references:
Oeis - Multiply-Perfect Numbers
MathWorld - Multiperfect Number
"Moxham (2000) found the largest known multiperfect number, approximately equal to 7.3×10^(1345), on Feb. 13, 2000."
The "The Multiply Perfect Numbers Page" has a gzipped file from 2014 with 5311 MPNs, as of 2018-01-07 no new numbers have been discovered.
answered Jul 21 at 7:59
Rob
346112
See Wikipedia's k-perfect numbers:
In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.
For a given natural number k, a number n is called k-perfect (or k-fold perfect) if and only if the sum of all positive divisors of n (the divisor function, Ã(n)) is equal to kn; a number is thus perfect if and only if it is 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number
$$
small
beginarrayrrl \
k & textSmallest k-perfect number & \
hline \
2 & 6 & 2 × 3 \
3 & 120 & 2^3 × 3 × 5 \
4 & 30240 & 2^5 × 3^3 × 5 × 7 \
5 & 14182439040 & 2^7 × 3^4 × 5 × 7 × 11^2 × 17 × 19 \
6 & text(21 digits) , 154345556085770649600 & 2^15 × 3^5 × 5^2 × 7^2 × 11 × 13 × 17 × 19 × 31 × 43 × 257 \
7 & ... & \
endarray
$$
See the Wikipedia page linked above for k-perfect numbers up to 11, they are too lengthy to fit on Stack Exchange pages. It's reasonable to expect that there are more but they haven't been discovered yet.
Other references:
Oeis - Multiply-Perfect Numbers
MathWorld - Multiperfect Number
"Moxham (2000) found the largest known multiperfect number, approximately equal to 7.3×10^(1345), on Feb. 13, 2000."
The "The Multiply Perfect Numbers Page" has a gzipped file from 2014 with 5311 MPNs, as of 2018-01-07 no new numbers have been discovered.
answered Jul 21 at 7:59
Rob
346112
edited Jul 21 at 8:30
answered Jul 21 at 7:59
Rob
346112
answered Jul 21 at 7:59
Rob
346112
answered Jul 21 at 7:59
Rob
346112
346112
add a comment |Â
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what is $sigma(n)$?
– uniquesolution
Jul 21 at 6:26
the sum of divisors function
– Tyler Litch
Jul 21 at 6:27
1
From the least numbers $n$ up to $k=11$ on Wikipedia, there seems to be an approximately exponential growth in the number of digits, very roughly $log npropto2.5^k$. From Grönwall's theorem, $$ limsup_nrightarrowinftyfracsigma(n)n,log log n=e^gamma;, $$ we might expect $log nproptomathrm e^k$.
– joriki
Jul 21 at 6:38
2
Nobody knows whether there is, for every $k>2$, an integer $n$ for which $sigma(n)=kn$.
– Gerry Myerson
Jul 21 at 7:15
Do we expect that there is? Why or why not?
– Tyler Litch
Jul 21 at 7:35