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