A divisor function with an apparent unique fixed point at 392

[binbots]

For a positive integer N, define


W(N) = sum over divisors d of N with d < sqrt(N) of (N/d - d)


and


E(N) = W(N) - (N - 1).


A few easy observations:

• E(p) = 0 for primes p.
• E(p²) = 0 for prime squares.
• If N = pq with primes p < q, then E(N) = q - p.

For example:


E(15) = 2
E(21) = 4
E(35) = 2


While experimenting with this function, I noticed an unexpected fixed point:


E(392) = 392.


The divisors of 392 below sqrt(392) are:


1, 2, 4, 7, 8, 14


and


W(392)
= (392-1) + (196-2) + (98-4) + (56-7) + (49-8) + (28-14)
= 783


so


E(392) = 783 - 391 = 392.


For nonsquare N, let


S(N) = sum of divisors below sqrt(N)


and


R(N) = sum of reciprocals of divisors below sqrt(N).


Then E(N) = N is equivalent to


R(N) - 2 = (S(N) - 1)/N.


I have checked all N <= 50,000,000 and found no other solutions to E(N) = N.


My questions are:

1. Is this function (or an equivalent form) already known?
2. Is there any known divisor-theoretic reason why 392 satisfies E(N) = N?
3. Does the reformulation
   R(N) - 2 = (S(N) - 1)/N
   suggest a route toward proving or disproving uniqueness?

I'm not claiming uniqueness, only that 392 appears to be the only example found so far computationally.

 

[fresh_42]

Looks funny, although a bit complicated.

I don't remember such a function. On the other side, there are quite a few number theoretical functions I don't have on my screen.

Zahlentheoretische Funktion – Wikipedia

de.wikipedia.org

Arithmetic function - Wikipedia

en.wikipedia.org

It would certainly help if you could rewrite your functions as an expression of some of these. WA also has some interesting properties,

392 - Wolfram|Alpha

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

www.wolframalpha.com

e.g., [imath] 392=14^2+14^2. [/imath]

 

[fresh_42]

My questions are:

1. Is this function (or an equivalent form) already known?

I don't think so, since your functions have an asymmetry in case [imath] N [/imath] is a square number. The divisor square root doesn't occur in either of your summations.

1. Is there any known divisor-theoretic reason why 392 satisfies E(N) = N?

See the answer to the previous question.

1. Does the reformulation
   R(N) - 2 = (S(N) - 1)/N
   suggest a route toward proving or disproving uniqueness?

We can only know if we saw a proof. Personally, I think, [imath] \sigma(n)=W(n)+2S(n) [/imath] is more helpful, but we have the problem again if [imath] \sqrt{N} [/imath] is an integer.

Your claim that [imath] N=392 [/imath] is the only solution to [imath] E(N)=N [/imath] can be rewritten as: [imath] N=392 [/imath] is the only solution to (if [imath] N [/imath] isn't a perfect square):
[math] \sigma(N)=2\left(N+\sum_{\substack{d|N\\d<\sqrt{N}}}d\right) -1 [/math]
From here on you can operate with known functions and the prime representation of [imath] N. [/imath]

 

[fresh_42]

If you write
[math] W(n)=\sum_{\substack{d|n\\d\ge\sqrt{n}}}d-\sum_{\substack{d|n\\d\le\sqrt{n}}}d [/math]then the difficulty becomes obvious: We have an additive equation, but separated the summation by a multiplicative rule. This mixture is always difficult to handle.