[math-fun] Progress in k-imperfect numbers
I'm including Michel Marcus because (a) I don't know if he's on this mailing list, and (b) he has helped me a lot in this research. The search for k-perfect numbers is well-known. We consider the function sigma(n), the sum of the divisors of n. Because sigma(n) >= n (with equality only when n = 1), it makes sense to look for numbers n such that sigma(n) = kn for some k; such a number is called k-perfect. (The "classical" perfect numbers are 2-perfect in this terminology.) The k-perfect numbers, for all indices k, are presented at OEIS sequence A007691. The following variation is due to Greg Martin, who presented it at the Western Number Theory Conference at Asilomar in 1999; it is "Western Number Theory problem 99:08". (See Myerson's compendium of WNT problems at https://www.math.colostate.edu/~achter/wntc/problems/problems2001.pdf .) Douglas Iannucci made some progress on the problem in his 2006 paper -- see http://math.colgate.edu/~integers/g41/g41.pdf . I'm using Iannucci's notation and nomenclature. Suppose d is a divisor of n. Consider the number of prime factors of the codivisor n/d, counting multiplicity. Call this the index of d in n. For example, the index of 2 in 12 is 2, because 12 = 2*(2*3). The index of 10 in 1000 is 4, because 1000 = 10*2*2*5*5. Now compute a _weighted_ sum of the divisors of n, where the weight of a divisor is 1 if the divisor has even index, and -1 if its index is odd. Call this weighted sum rho(n). For example, rho(12) = 12 - 6 - 4 + 3 + 2 - 1 = 6. Note that rho(n) <= n (with equality only when n = 1), so it makes sense to look for numbers n such that k*rho(n) = n for some k; Iannucci calls such a number k-imperfect. (I personally would have preferred "k-contraperfect", but that ship has sailed.) Because 12 = 2*rho(12), 12 is 2-imperfect. The function rho(n) is multiplicative, with rho(p^e) = p^e - p^(e-1) + ... +/- 1; the sign of the trailing unit depends on the parity of e. Like k-perfect numbers, there are lots of known examples of k-imperfect numbers. Many of these are listed at https://oeis.org/A127724 . Because they get big so quickly, we soon lose track of whether they are consecutive; we are pretty sure we know the smallest 50, but after that, the gaps have not been searched exhaustively. About 1800 examples are known; part of my recent research has added around 200 new ones. While k-perfect numbers are known for all 1 <= k <= 11, for k-imperfect numbers, we only had examples of k = 1, 2, 3, or 4 ... until earlier today, when I found one with k = 5. Martin and Iannucci had only found values of k up to 3, but many examples with k = 4 were subsequently found by Corneth, Johnson, Lelechenko, and Marcus (Michel Marcus found more than a thousand). We didn't know for sure if there were any 5-imperfect numbers until this afternoon. The smallest known 4-imperfect number is 993803899780063855042560 (24 digits). My discovery today is so far the only known 5-imperfect number; it has 208 digits, and the exact value is 1947793410288108579327587698415272737289992039373107522449638016140636142596276017072442826236838486130285072853813458640948347868163450516845165654669897619318450994951647517899051394662400000000000000000000. Michel Marcus and I have both checked it (he with a package he wrote in Pari, I with a complete cludge in Emacs Lisp). I would welcome additional confirmation. I have just emerged from a "dry valley" in my search space, and expect to find at least a few more 5-imperfect numbers in the next few days, but I thought I shouldn't let the occasion of the discovery of the first example go unmentioned.
Hi Allan, Congrats on your BigInt discovery, and nice write up. Re: "imperfect", another possibility would be "infraperfect". But the phrase starting "it makes sense..." did not make sense to me, and it could probably be replaced by saying "some k might exist such that sigma(n) = k*n; such a number..." Whatever does or does not make sense is subjective to a person's own capabilities and interests. Sure, I would like to help you by double checking your calculation. Unfortunately, It doesn't make sense for me to try to do so, because I haven't invested in the necessary skills. In lieu of just shutting up, let me point out another minor issue. Per what I could find in either reference, it doesn't make sense to me why this particular linear functional would be better than any other. Is it, as I fear, mainly another example of //argumentum ab auctoritate// in number theory? This question is not to rain on your parade, or diminish your accomplishment, but simply to say that it could possibly "make sense" to diversify the divisor analysis. For example, we could add odds and subtract evens. I calculate quickly that: Flatten[Position[IntegerQ[Total[ Divisors[#] /. x_ /; EvenQ[x] :> -x ]/# ] & /@ Range[100000], True]] Out[]:= {1, 60, 728, 6960, 60512, 97152} Would these "extraperfect" numbers be of any interest, even to OEIS? Cheers, Brad On Tue, Feb 11, 2020 at 10:04 PM Allan Wechsler <acwacw@gmail.com> wrote:
I'm including Michel Marcus because (a) I don't know if he's on this mailing list, and (b) he has helped me a lot in this research.
The search for k-perfect numbers is well-known. We consider the function sigma(n), the sum of the divisors of n. Because sigma(n) >= n (with equality only when n = 1), it makes sense to look for numbers n such that sigma(n) = kn for some k; such a number is called k-perfect. (The "classical" perfect numbers are 2-perfect in this terminology.) The k-perfect numbers, for all indices k, are presented at OEIS sequence A007691.
The following variation is due to Greg Martin, who presented it at the Western Number Theory Conference at Asilomar in 1999; it is "Western Number Theory problem 99:08". (See Myerson's compendium of WNT problems at https://www.math.colostate.edu/~achter/wntc/problems/problems2001.pdf .)
Douglas Iannucci made some progress on the problem in his 2006 paper -- see http://math.colgate.edu/~integers/g41/g41.pdf . I'm using Iannucci's notation and nomenclature.
Suppose d is a divisor of n. Consider the number of prime factors of the codivisor n/d, counting multiplicity. Call this the index of d in n. For example, the index of 2 in 12 is 2, because 12 = 2*(2*3). The index of 10 in 1000 is 4, because 1000 = 10*2*2*5*5. Now compute a _weighted_ sum of the divisors of n, where the weight of a divisor is 1 if the divisor has even index, and -1 if its index is odd. Call this weighted sum rho(n). For example, rho(12) = 12 - 6 - 4 + 3 + 2 - 1 = 6. Note that rho(n) <= n (with equality only when n = 1), so it makes sense to look for numbers n such that k*rho(n) = n for some k; Iannucci calls such a number k-imperfect. (I personally would have preferred "k-contraperfect", but that ship has sailed.) Because 12 = 2*rho(12), 12 is 2-imperfect.
The function rho(n) is multiplicative, with rho(p^e) = p^e - p^(e-1) + ... +/- 1; the sign of the trailing unit depends on the parity of e.
Like k-perfect numbers, there are lots of known examples of k-imperfect numbers. Many of these are listed at https://oeis.org/A127724 . Because they get big so quickly, we soon lose track of whether they are consecutive; we are pretty sure we know the smallest 50, but after that, the gaps have not been searched exhaustively. About 1800 examples are known; part of my recent research has added around 200 new ones.
While k-perfect numbers are known for all 1 <= k <= 11, for k-imperfect numbers, we only had examples of k = 1, 2, 3, or 4 ... until earlier today, when I found one with k = 5. Martin and Iannucci had only found values of k up to 3, but many examples with k = 4 were subsequently found by Corneth, Johnson, Lelechenko, and Marcus (Michel Marcus found more than a thousand). We didn't know for sure if there were any 5-imperfect numbers until this afternoon.
The smallest known 4-imperfect number is 993803899780063855042560 (24 digits). My discovery today is so far the only known 5-imperfect number; it has 208 digits, and the exact value is 1947793410288108579327587698415272737289992039373107522449638016140636142596276017072442826236838486130285072853813458640948347868163450516845165654669897619318450994951647517899051394662400000000000000000000. Michel Marcus and I have both checked it (he with a package he wrote in Pari, I with a complete cludge in Emacs Lisp). I would welcome additional confirmation.
I have just emerged from a "dry valley" in my search space, and expect to find at least a few more 5-imperfect numbers in the next few days, but I thought I shouldn't let the occasion of the discovery of the first example go unmentioned. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Hi, Brad, sorry I didn't get back to you on this topic sooner. I agree that my "makes sense" trope was a little bit incoherent. What I meant was that, historically, a bunch of folks have considered it interesting when n divides sigma(n), and have so considered it since classical times (for k=2 only), and since the Enlightenment (for higher values of k). Is the puzzle worth the attention? I can't judge -- all I know is that since I've started looking at it, it has interested _me_. I don't think it's especially deep or profound, but the chase is attractive in and of itself, and this is probably purely a matter of taste. The related problem, with rho(n) dividing n instead of n dividing sigma(n), doesn't have the same pedigree; nobody seems to have thought of it before Martin posed it at the Asilomar conference. Since then, a few people have joined this mutant chase -- maybe a dozen people have looked at it. We're just looking for solutions to an equation in a weird multiplicative function. Is it worth doing? I don't know; I'm enjoying it, and the techniques I learned while working on the old problem let me make a breakthrough in the new one. The function that you proposed seems reasonably interesting to me, as well. I hadn't thought of it (and I have spent some time considering alternative multiplicative functions). But it's at least obvious enough to appear in the OEIS -- see https://oeis.org/A002129. Glaisher noted the function in 1907; it has some connection with representing a number as the some of a small even number of squares. He called it zeta(n). I don't think anybody has investigated cases where n divides zeta(n) or vice versa. There are some examples, though. When n=728, zeta(n) is -1456, so 728 is "Kleeperfect" with order negative 2. The usual questions would apply: for example, are there any odd Kleeperfect numbers? The fact that zeta(n) is negative for even n adds a new interesting wrinkle. Perhaps some math-funsters might be interested in chasing this particular rabbit; for the moment I am satisfied with my own project. On Wed, Feb 12, 2020 at 2:58 AM Brad Klee <bradklee@gmail.com> wrote:
Hi Allan,
Congrats on your BigInt discovery, and nice write up. Re: "imperfect", another possibility would be "infraperfect".
But the phrase starting "it makes sense..." did not make sense to me, and it could probably be replaced by saying "some k might exist such that sigma(n) = k*n; such a number..."
Whatever does or does not make sense is subjective to a person's own capabilities and interests. Sure, I would like to help you by double checking your calculation. Unfortunately, It doesn't make sense for me to try to do so, because I haven't invested in the necessary skills.
In lieu of just shutting up, let me point out another minor issue. Per what I could find in either reference, it doesn't make sense to me why this particular linear functional would be better than any other. Is it, as I fear, mainly another example of //argumentum ab auctoritate// in number theory?
This question is not to rain on your parade, or diminish your accomplishment, but simply to say that it could possibly "make sense" to diversify the divisor analysis. For example, we could add odds and subtract evens. I calculate quickly that:
Flatten[Position[IntegerQ[Total[ Divisors[#] /. x_ /; EvenQ[x] :> -x ]/# ] & /@ Range[100000], True]]
Out[]:= {1, 60, 728, 6960, 60512, 97152}
Would these "extraperfect" numbers be of any interest, even to OEIS?
Cheers,
Brad
On Tue, Feb 11, 2020 at 10:04 PM Allan Wechsler <acwacw@gmail.com> wrote:
I'm including Michel Marcus because (a) I don't know if he's on this mailing list, and (b) he has helped me a lot in this research.
The search for k-perfect numbers is well-known. We consider the function sigma(n), the sum of the divisors of n. Because sigma(n) >= n (with equality only when n = 1), it makes sense to look for numbers n such that sigma(n) = kn for some k; such a number is called k-perfect. (The "classical" perfect numbers are 2-perfect in this terminology.) The k-perfect numbers, for all indices k, are presented at OEIS sequence A007691.
The following variation is due to Greg Martin, who presented it at the Western Number Theory Conference at Asilomar in 1999; it is "Western
Number
Theory problem 99:08". (See Myerson's compendium of WNT problems at https://www.math.colostate.edu/~achter/wntc/problems/problems2001.pdf .)
Douglas Iannucci made some progress on the problem in his 2006 paper -- see http://math.colgate.edu/~integers/g41/g41.pdf . I'm using Iannucci's notation and nomenclature.
Suppose d is a divisor of n. Consider the number of prime factors of the codivisor n/d, counting multiplicity. Call this the index of d in n. For example, the index of 2 in 12 is 2, because 12 = 2*(2*3). The index of 10 in 1000 is 4, because 1000 = 10*2*2*5*5. Now compute a _weighted_ sum of the divisors of n, where the weight of a divisor is 1 if the divisor has even index, and -1 if its index is odd. Call this weighted sum rho(n). For example, rho(12) = 12 - 6 - 4 + 3 + 2 - 1 = 6. Note that rho(n) <= n (with equality only when n = 1), so it makes sense to look for numbers n such that k*rho(n) = n for some k; Iannucci calls such a number k-imperfect. (I personally would have preferred "k-contraperfect", but that ship has sailed.) Because 12 = 2*rho(12), 12 is 2-imperfect.
The function rho(n) is multiplicative, with rho(p^e) = p^e - p^(e-1) + ... +/- 1; the sign of the trailing unit depends on the parity of e.
Like k-perfect numbers, there are lots of known examples of k-imperfect numbers. Many of these are listed at https://oeis.org/A127724 . Because they get big so quickly, we soon lose track of whether they are consecutive; we are pretty sure we know the smallest 50, but after that, the gaps have not been searched exhaustively. About 1800 examples are known; part of my recent research has added around 200 new ones.
While k-perfect numbers are known for all 1 <= k <= 11, for k-imperfect numbers, we only had examples of k = 1, 2, 3, or 4 ... until earlier today, when I found one with k = 5. Martin and Iannucci had only found values of k up to 3, but many examples with k = 4 were subsequently found by Corneth, Johnson, Lelechenko, and Marcus (Michel Marcus found more than a thousand). We didn't know for sure if there were any 5-imperfect numbers until this afternoon.
The smallest known 4-imperfect number is 993803899780063855042560 (24 digits). My discovery today is so far the only known 5-imperfect number; it has 208 digits, and the exact value is
1947793410288108579327587698415272737289992039373107522449638016140636142596276017072442826236838486130285072853813458640948347868163450516845165654669897619318450994951647517899051394662400000000000000000000.
Michel Marcus and I have both checked it (he with a package he wrote in Pari, I with a complete cludge in Emacs Lisp). I would welcome additional confirmation.
I have just emerged from a "dry valley" in my search space, and expect to find at least a few more 5-imperfect numbers in the next few days, but I thought I shouldn't let the occasion of the discovery of the first example go unmentioned. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
When your calculations are right and / or difficult, it seems like you can always expect someone to complain about your language—a priggish but pervasive learned behavior of science. It could help, *occasionally*. Hopefully the appellation “kleeperfect” does not catch on, even though it is kind of funny as an oxymoron. “Glaisherperfect” would be a better patronymic. The name Glaisher is a good one in England (twice FRS puts it in the same class as Penrose). The name Klee in America is neither renowned nor authoritative, except for the well-liked Victor Klee (of no relation to me). The other proposal to call them “extraperfect” numbers uses "extra" in the sense of “extraneous”. Then again, someone could likely read this prefix in the sense of “extraordinary”, so I don’t know what we would call them. Another question about these who-really-knows-what-they-are-numbers is whether or not one of them is ever also a perfect number. With an answer, we could call them either "occasionally perfect numbers" or "never perfect numbers". I calculated up to 10^6, and found one more: 728,000, which again is not perfect. Another congrats on your discovery, glad you are happy with it. That part makes plenty of sense, because it is exciting and worthwhile to discover new species. --Brad
On Feb 22, 2020, at 9:07 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Hi, Brad, sorry I didn't get back to you on this topic sooner. I agree that
my "makes sense" trope was a little bit incoherent. What I meant was that, historically, a bunch of folks have considered it interesting when n divides sigma(n), and have so considered it since classical times (for k=2 only), and since the Enlightenment (for higher values of k). Is the puzzle worth the attention? I can't judge -- all I know is that since I've started looking at it, it has interested _me_. I don't think it's especially deep or profound, but the chase is attractive in and of itself, and this is probably purely a matter of taste.
The related problem, with rho(n) dividing n instead of n dividing sigma(n), doesn't have the same pedigree; nobody seems to have thought of it before Martin posed it at the Asilomar conference. Since then, a few people have joined this mutant chase -- maybe a dozen people have looked at it. We're just looking for solutions to an equation in a weird multiplicative function. Is it worth doing? I don't know; I'm enjoying it, and the techniques I learned while working on the old problem let me make a breakthrough in the new one.
The function that you proposed seems reasonably interesting to me, as well. I hadn't thought of it (and I have spent some time considering alternative multiplicative functions). But it's at least obvious enough to appear in the OEIS -- see https://oeis.org/A002129. Glaisher noted the function in 1907; it has some connection with representing a number as the some of a small even number of squares. He called it zeta(n). I don't think anybody has investigated cases where n divides zeta(n) or vice versa. There are some examples, though. When n=728, zeta(n) is -1456, so 728 is "Kleeperfect" with order negative 2. The usual questions would apply: for example, are there any odd Kleeperfect numbers? The fact that zeta(n) is negative for even n adds a new interesting wrinkle. Perhaps some math-funsters might be interested in chasing this particular rabbit; for the moment I am satisfied with my own project.
On Wed, Feb 12, 2020 at 2:58 AM Brad Klee <bradklee@gmail.com> wrote:
Hi Allan,
Congrats on your BigInt discovery, and nice write up. Re: "imperfect", another possibility would be "infraperfect".
But the phrase starting "it makes sense..." did not make sense to me, and it could probably be replaced by saying "some k might exist such that sigma(n) = k*n; such a number..."
Whatever does or does not make sense is subjective to a person's own capabilities and interests. Sure, I would like to help you by double checking your calculation. Unfortunately, It doesn't make sense for me to try to do so, because I haven't invested in the necessary skills.
In lieu of just shutting up, let me point out another minor issue. Per what I could find in either reference, it doesn't make sense to me why this particular linear functional would be better than any other. Is it, as I fear, mainly another example of //argumentum ab auctoritate// in number theory?
This question is not to rain on your parade, or diminish your accomplishment, but simply to say that it could possibly "make sense" to diversify the divisor analysis. For example, we could add odds and subtract evens. I calculate quickly that:
Flatten[Position[IntegerQ[Total[ Divisors[#] /. x_ /; EvenQ[x] :> -x ]/# ] & /@ Range[100000], True]]
Out[]:= {1, 60, 728, 6960, 60512, 97152}
Would these "extraperfect" numbers be of any interest, even to OEIS?
Cheers,
Brad
On Tue, Feb 11, 2020 at 10:04 PM Allan Wechsler <acwacw@gmail.com> wrote:
I'm including Michel Marcus because (a) I don't know if he's on this mailing list, and (b) he has helped me a lot in this research.
The search for k-perfect numbers is well-known. We consider the function sigma(n), the sum of the divisors of n. Because sigma(n) >= n (with equality only when n = 1), it makes sense to look for numbers n such that sigma(n) = kn for some k; such a number is called k-perfect. (The "classical" perfect numbers are 2-perfect in this terminology.) The k-perfect numbers, for all indices k, are presented at OEIS sequence A007691.
The following variation is due to Greg Martin, who presented it at the Western Number Theory Conference at Asilomar in 1999; it is "Western Number Theory problem 99:08". (See Myerson's compendium of WNT problems at https://www.math.colostate.edu/~achter/wntc/problems/problems2001.pdf .)
Douglas Iannucci made some progress on the problem in his 2006 paper -- see http://math.colgate.edu/~integers/g41/g41.pdf . I'm using Iannucci's notation and nomenclature.
Suppose d is a divisor of n. Consider the number of prime factors of the codivisor n/d, counting multiplicity. Call this the index of d in n. For example, the index of 2 in 12 is 2, because 12 = 2*(2*3). The index of 10 in 1000 is 4, because 1000 = 10*2*2*5*5. Now compute a _weighted_ sum of the divisors of n, where the weight of a divisor is 1 if the divisor has even index, and -1 if its index is odd. Call this weighted sum rho(n). For example, rho(12) = 12 - 6 - 4 + 3 + 2 - 1 = 6. Note that rho(n) <= n (with equality only when n = 1), so it makes sense to look for numbers n such that k*rho(n) = n for some k; Iannucci calls such a number k-imperfect. (I personally would have preferred "k-contraperfect", but that ship has sailed.) Because 12 = 2*rho(12), 12 is 2-imperfect.
The function rho(n) is multiplicative, with rho(p^e) = p^e - p^(e-1) + ... +/- 1; the sign of the trailing unit depends on the parity of e.
Like k-perfect numbers, there are lots of known examples of k-imperfect numbers. Many of these are listed at https://oeis.org/A127724 . Because they get big so quickly, we soon lose track of whether they are consecutive; we are pretty sure we know the smallest 50, but after that, the gaps have not been searched exhaustively. About 1800 examples are known; part of my recent research has added around 200 new ones.
While k-perfect numbers are known for all 1 <= k <= 11, for k-imperfect numbers, we only had examples of k = 1, 2, 3, or 4 ... until earlier today, when I found one with k = 5. Martin and Iannucci had only found values of k up to 3, but many examples with k = 4 were subsequently found by Corneth, Johnson, Lelechenko, and Marcus (Michel Marcus found more than a thousand). We didn't know for sure if there were any 5-imperfect numbers until this afternoon.
The smallest known 4-imperfect number is 993803899780063855042560 (24 digits). My discovery today is so far the only known 5-imperfect number; it has 208 digits, and the exact value is
1947793410288108579327587698415272737289992039373107522449638016140636142596276017072442826236838486130285072853813458640948347868163450516845165654669897619318450994951647517899051394662400000000000000000000.
Michel Marcus and I have both checked it (he with a package he wrote in Pari, I with a complete cludge in Emacs Lisp). I would welcome additional confirmation.
I have just emerged from a "dry valley" in my search space, and expect to find at least a few more 5-imperfect numbers in the next few days, but I thought I shouldn't let the occasion of the discovery of the first example go unmentioned. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
You don't mention Paul Klee, whose work I've admired since childhood. I'd like to visit the Zentrum Paul Klee in Bern sometime. I can confirm 728,000, index -3. I find at least three other examples in the range you specify: 60 (-2), 2,920 (-3), and 60,512 (-2). I can't be sure I have all of them below 10^6, because my heuristics are not exhaustive. But just now, I also turned up 1,900,080, of index -4. As I said, though, these are not calling to me at the moment, so somebody else should name them. On Sun, Feb 23, 2020 at 12:24 AM Brad Klee <bradklee@gmail.com> wrote:
When your calculations are right and / or difficult, it seems like you can always expect someone to complain about your language—a priggish but pervasive learned behavior of science. It could help, *occasionally*.
Hopefully the appellation “kleeperfect” does not catch on, even though it is kind of funny as an oxymoron. “Glaisherperfect” would be a better patronymic. The name Glaisher is a good one in England (twice FRS puts it in the same class as Penrose). The name Klee in America is neither renowned nor authoritative, except for the well-liked Victor Klee (of no relation to me).
The other proposal to call them “extraperfect” numbers uses "extra" in the sense of “extraneous”. Then again, someone could likely read this prefix in the sense of “extraordinary”, so I don’t know what we would call them.
Another question about these who-really-knows-what-they-are-numbers is whether or not one of them is ever also a perfect number. With an answer, we could call them either "occasionally perfect numbers" or "never perfect numbers".
I calculated up to 10^6, and found one more: 728,000, which again is not perfect.
Another congrats on your discovery, glad you are happy with it. That part makes plenty of sense, because it is exciting and worthwhile to discover new species.
--Brad
On Feb 22, 2020, at 9:07 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Hi, Brad, sorry I didn't get back to you on this topic sooner. I agree that
my "makes sense" trope was a little bit incoherent. What I meant was that, historically, a bunch of folks have considered it interesting when n divides sigma(n), and have so considered it since classical times (for k=2 only), and since the Enlightenment (for higher values of k). Is the puzzle worth the attention? I can't judge -- all I know is that since I've started looking at it, it has interested _me_. I don't think it's especially deep or profound, but the chase is attractive in and of itself, and this is probably purely a matter of taste.
The related problem, with rho(n) dividing n instead of n dividing sigma(n), doesn't have the same pedigree; nobody seems to have thought of it before Martin posed it at the Asilomar conference. Since then, a few people have joined this mutant chase -- maybe a dozen people have looked at it. We're just looking for solutions to an equation in a weird multiplicative function. Is it worth doing? I don't know; I'm enjoying it, and the techniques I learned while working on the old problem let me make a breakthrough in the new one.
The function that you proposed seems reasonably interesting to me, as well. I hadn't thought of it (and I have spent some time considering alternative multiplicative functions). But it's at least obvious enough to appear in the OEIS -- see https://oeis.org/A002129. Glaisher noted the function in 1907; it has some connection with representing a number as the some of a small even number of squares. He called it zeta(n). I don't think anybody has investigated cases where n divides zeta(n) or vice versa. There are some examples, though. When n=728, zeta(n) is -1456, so 728 is "Kleeperfect" with order negative 2. The usual questions would apply: for example, are there any odd Kleeperfect numbers? The fact that zeta(n) is negative for even n adds a new interesting wrinkle. Perhaps some math-funsters might be interested in chasing this particular rabbit; for the moment I am satisfied with my own project.
On Wed, Feb 12, 2020 at 2:58 AM Brad Klee <bradklee@gmail.com> wrote:
Hi Allan,
Congrats on your BigInt discovery, and nice write up. Re: "imperfect", another possibility would be "infraperfect".
But the phrase starting "it makes sense..." did not make sense to me, and it could probably be replaced by saying "some k might exist such that sigma(n) = k*n; such a number..."
Whatever does or does not make sense is subjective to a person's own capabilities and interests. Sure, I would like to help you by double checking your calculation. Unfortunately, It doesn't make sense for me to try to do so, because I haven't invested in the necessary skills.
In lieu of just shutting up, let me point out another minor issue. Per what I could find in either reference, it doesn't make sense to me why this particular linear functional would be better than any other. Is it, as I fear, mainly another example of //argumentum ab auctoritate// in number theory?
This question is not to rain on your parade, or diminish your accomplishment, but simply to say that it could possibly "make sense" to diversify the divisor analysis. For example, we could add odds and subtract evens. I calculate quickly that:
Flatten[Position[IntegerQ[Total[ Divisors[#] /. x_ /; EvenQ[x] :> -x ]/# ] & /@ Range[100000], True]]
Out[]:= {1, 60, 728, 6960, 60512, 97152}
Would these "extraperfect" numbers be of any interest, even to OEIS?
Cheers,
Brad
On Tue, Feb 11, 2020 at 10:04 PM Allan Wechsler <acwacw@gmail.com> wrote:
I'm including Michel Marcus because (a) I don't know if he's on this mailing list, and (b) he has helped me a lot in this research.
The search for k-perfect numbers is well-known. We consider the function sigma(n), the sum of the divisors of n. Because sigma(n) >= n (with equality only when n = 1), it makes sense to look for numbers n such that sigma(n) = kn for some k; such a number is called k-perfect. (The "classical" perfect numbers are 2-perfect in this terminology.) The k-perfect numbers, for all indices k, are presented at OEIS sequence A007691.
The following variation is due to Greg Martin, who presented it at the Western Number Theory Conference at Asilomar in 1999; it is "Western Number Theory problem 99:08". (See Myerson's compendium of WNT problems at https://www.math.colostate.edu/~achter/wntc/problems/problems2001.pdf .)
Douglas Iannucci made some progress on the problem in his 2006 paper -- see http://math.colgate.edu/~integers/g41/g41.pdf . I'm using Iannucci's notation and nomenclature.
Suppose d is a divisor of n. Consider the number of prime factors of the codivisor n/d, counting multiplicity. Call this the index of d in n. For example, the index of 2 in 12 is 2, because 12 = 2*(2*3). The index of 10 in 1000 is 4, because 1000 = 10*2*2*5*5. Now compute a _weighted_ sum of the divisors of n, where the weight of a divisor is 1 if the divisor has even index, and -1 if its index is odd. Call this weighted sum rho(n). For example, rho(12) = 12 - 6 - 4 + 3 + 2 - 1 = 6. Note that rho(n) <= n (with equality only when n = 1), so it makes sense to look for numbers n such that k*rho(n) = n for some k; Iannucci calls such a number k-imperfect. (I personally would have preferred "k-contraperfect", but that ship has sailed.) Because 12 = 2*rho(12), 12 is 2-imperfect.
The function rho(n) is multiplicative, with rho(p^e) = p^e - p^(e-1) + ... +/- 1; the sign of the trailing unit depends on the parity of e.
Like k-perfect numbers, there are lots of known examples of k-imperfect numbers. Many of these are listed at https://oeis.org/A127724 . Because they get big so quickly, we soon lose track of whether they are consecutive; we are pretty sure we know the smallest 50, but after that, the gaps have not been searched exhaustively. About 1800 examples are known; part of my recent research has added around 200 new ones.
While k-perfect numbers are known for all 1 <= k <= 11, for k-imperfect numbers, we only had examples of k = 1, 2, 3, or 4 ... until earlier today, when I found one with k = 5. Martin and Iannucci had only found values of k up to 3, but many examples with k = 4 were subsequently found by Corneth, Johnson, Lelechenko, and Marcus (Michel Marcus found more than a thousand). We didn't know for sure if there were any 5-imperfect numbers until this afternoon.
The smallest known 4-imperfect number is 993803899780063855042560 (24 digits). My discovery today is so far the only known 5-imperfect number; it has 208 digits, and the exact value is
1947793410288108579327587698415272737289992039373107522449638016140636142596276017072442826236838486130285072853813458640948347868163450516845165654669897619318450994951647517899051394662400000000000000000000.
Michel Marcus and I have both checked it (he with a package he wrote in Pari, I with a complete cludge in Emacs Lisp). I would welcome additional confirmation.
I have just emerged from a "dry valley" in my search space, and expect to find at least a few more 5-imperfect numbers in the next few days, but I thought I shouldn't let the occasion of the discovery of the first example go unmentioned. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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There’s also Steven Klee of the University of Washington. Jim On Sun, Feb 23, 2020 at 7:51 AM Allan Wechsler <acwacw@gmail.com> wrote:
You don't mention Paul Klee, whose work I've admired since childhood. I'd like to visit the Zentrum Paul Klee in Bern sometime.
I can confirm 728,000, index -3. I find at least three other examples in the range you specify: 60 (-2), 2,920 (-3), and 60,512 (-2). I can't be sure I have all of them below 10^6, because my heuristics are not exhaustive. But just now, I also turned up 1,900,080, of index -4. As I said, though, these are not calling to me at the moment, so somebody else should name them.
On Sun, Feb 23, 2020 at 12:24 AM Brad Klee <bradklee@gmail.com> wrote:
When your calculations are right and / or difficult, it seems like you can always expect someone to complain about your language—a priggish but pervasive learned behavior of science. It could help, *occasionally*.
Hopefully the appellation “kleeperfect” does not catch on, even though it is kind of funny as an oxymoron. “Glaisherperfect” would be a better patronymic. The name Glaisher is a good one in England (twice FRS puts it in the same class as Penrose). The name Klee in America is neither renowned nor authoritative, except for the well-liked Victor Klee (of no relation to me).
The other proposal to call them “extraperfect” numbers uses "extra" in the sense of “extraneous”. Then again, someone could likely read this prefix in the sense of “extraordinary”, so I don’t know what we would call them.
Another question about these who-really-knows-what-they-are-numbers is whether or not one of them is ever also a perfect number. With an answer, we could call them either "occasionally perfect numbers" or "never perfect numbers".
I calculated up to 10^6, and found one more: 728,000, which again is not perfect.
Another congrats on your discovery, glad you are happy with it. That part makes plenty of sense, because it is exciting and worthwhile to discover new species.
--Brad
On Feb 22, 2020, at 9:07 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Hi, Brad, sorry I didn't get back to you on this topic sooner. I agree that
my "makes sense" trope was a little bit incoherent. What I meant was that, historically, a bunch of folks have considered it interesting when n divides sigma(n), and have so considered it since classical times (for k=2 only), and since the Enlightenment (for higher values of k). Is the puzzle worth the attention? I can't judge -- all I know is that since I've started looking at it, it has interested _me_. I don't think it's especially deep or profound, but the chase is attractive in and of itself, and this is probably purely a matter of taste.
The related problem, with rho(n) dividing n instead of n dividing sigma(n), doesn't have the same pedigree; nobody seems to have thought of it before Martin posed it at the Asilomar conference. Since then, a few people have joined this mutant chase -- maybe a dozen people have looked at it. We're just looking for solutions to an equation in a weird multiplicative function. Is it worth doing? I don't know; I'm enjoying it, and the techniques I learned while working on the old problem let me make a breakthrough in the new one.
The function that you proposed seems reasonably interesting to me, as well. I hadn't thought of it (and I have spent some time considering alternative multiplicative functions). But it's at least obvious enough to appear in the OEIS -- see https://oeis.org/A002129. Glaisher noted the function in 1907; it has some connection with representing a number as the some of a small even number of squares. He called it zeta(n). I don't think anybody has investigated cases where n divides zeta(n) or vice versa. There are some examples, though. When n=728, zeta(n) is -1456, so 728 is "Kleeperfect" with order negative 2. The usual questions would apply: for example, are there any odd Kleeperfect numbers? The fact that zeta(n) is negative for even n adds a new interesting wrinkle. Perhaps some math-funsters might be interested in chasing this particular rabbit; for the moment I am satisfied with my own project.
On Wed, Feb 12, 2020 at 2:58 AM Brad Klee <bradklee@gmail.com> wrote:
Hi Allan,
Congrats on your BigInt discovery, and nice write up. Re: "imperfect", another possibility would be "infraperfect".
But the phrase starting "it makes sense..." did not make sense to me, and it could probably be replaced by saying "some k might exist such that sigma(n) = k*n; such a number..."
Whatever does or does not make sense is subjective to a person's own capabilities and interests. Sure, I would like to help you by double checking your calculation. Unfortunately, It doesn't make sense for me to try to do so, because I haven't invested in the necessary skills.
In lieu of just shutting up, let me point out another minor issue. Per what I could find in either reference, it doesn't make sense to me why this particular linear functional would be better than any other. Is it, as I fear, mainly another example of //argumentum ab auctoritate// in number theory?
This question is not to rain on your parade, or diminish your accomplishment, but simply to say that it could possibly "make sense" to diversify the divisor analysis. For example, we could add odds and subtract evens. I calculate quickly that:
Flatten[Position[IntegerQ[Total[ Divisors[#] /. x_ /; EvenQ[x] :> -x ]/# ] & /@ Range[100000], True]]
Out[]:= {1, 60, 728, 6960, 60512, 97152}
Would these "extraperfect" numbers be of any interest, even to OEIS?
Cheers,
Brad
On Tue, Feb 11, 2020 at 10:04 PM Allan Wechsler <acwacw@gmail.com> wrote:
I'm including Michel Marcus because (a) I don't know if he's on this mailing list, and (b) he has helped me a lot in this research.
The search for k-perfect numbers is well-known. We consider the function sigma(n), the sum of the divisors of n. Because sigma(n) >= n (with equality only when n = 1), it makes sense to look for numbers n such that sigma(n) = kn for some k; such a number is called k-perfect. (The "classical" perfect numbers are 2-perfect in this terminology.) The k-perfect numbers, for all indices k, are presented at OEIS sequence A007691.
The following variation is due to Greg Martin, who presented it at the Western Number Theory Conference at Asilomar in 1999; it is "Western Number Theory problem 99:08". (See Myerson's compendium of WNT problems at
https://www.math.colostate.edu/~achter/wntc/problems/problems2001.pdf .)
Douglas Iannucci made some progress on the problem in his 2006 paper
-- see
http://math.colgate.edu/~integers/g41/g41.pdf . I'm using Iannucci's notation and nomenclature.
Suppose d is a divisor of n. Consider the number of prime factors of the codivisor n/d, counting multiplicity. Call this the index of d in n. For example, the index of 2 in 12 is 2, because 12 = 2*(2*3). The index of 10 in 1000 is 4, because 1000 = 10*2*2*5*5. Now compute a _weighted_ sum of the divisors of n, where the weight of a divisor is 1 if the divisor has even index, and -1 if its index is odd. Call this weighted sum rho(n). For example, rho(12) = 12 - 6 - 4 + 3 + 2 - 1 = 6. Note that rho(n) <= n (with equality only when n = 1), so it makes sense to look for numbers n such that k*rho(n) = n for some k; Iannucci calls such a number k-imperfect. (I personally would have preferred "k-contraperfect", but that ship has sailed.) Because 12 = 2*rho(12), 12 is 2-imperfect.
The function rho(n) is multiplicative, with rho(p^e) = p^e - p^(e-1)
... +/- 1; the sign of the trailing unit depends on the parity of e.
Like k-perfect numbers, there are lots of known examples of k-imperfect numbers. Many of these are listed at https://oeis.org/A127724 . Because they get big so quickly, we soon lose track of whether they are consecutive; we are pretty sure we know the smallest 50, but after that, the gaps have not been searched exhaustively. About 1800 examples are known; part of my recent research has added around 200 new ones.
While k-perfect numbers are known for all 1 <= k <= 11, for k-imperfect numbers, we only had examples of k = 1, 2, 3, or 4 ... until earlier today, when I found one with k = 5. Martin and Iannucci had only found values of k up to 3, but many examples with k = 4 were subsequently found by Corneth, Johnson, Lelechenko, and Marcus (Michel Marcus found more than a thousand). We didn't know for sure if there were any 5-imperfect numbers until this afternoon.
The smallest known 4-imperfect number is 993803899780063855042560 (24 digits). My discovery today is so far the only known 5-imperfect number; it has 208 digits, and the exact value is
1947793410288108579327587698415272737289992039373107522449638016140636142596276017072442826236838486130285072853813458640948347868163450516845165654669897619318450994951647517899051394662400000000000000000000.
Michel Marcus and I have both checked it (he with a package he wrote in Pari, I with a complete cludge in Emacs Lisp). I would welcome additional confirmation.
I have just emerged from a "dry valley" in my search space, and expect to find at least a few more 5-imperfect numbers in the next few days, but I thought I shouldn't let the occasion of the discovery of the first example go unmentioned. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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participants (3)
-
Allan Wechsler -
Brad Klee -
James Propp