A divisor “mirror imbalance” sequence I can’t identify

[binbots]

I’ve been experimenting with a divisor symmetry statistic and it produces a sequence I haven’t been able to locate in the literature or OEIS.


For an integer N, define:


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


Then define:


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


So E(N) measures how far the divisor structure deviates from what happens for primes.


Some observations:

• If N is prime, W(N) = N-1 and E(N) = 0
• If N = p^2 (prime squared), E(N) = 0
• If N = p*q with primes p < q, then E(N) = q - p
  (i.e., the “excess” equals the prime gap inside the semiprime)

So the statistic seems to measure something like a divisor “mirror imbalance” around sqrt(N).


Here are the first values of E(N) for N = 1 to 50:


0, 0, 0, 0, 0, 1, 0, 2, 0, 3, 0, 5, 0, 5, 2, 6, 0, 10, 0, 9, 4, 9, 0, 17, 0, 11, 6, 15, 0, 21, 0, 18, 8, 15, 2, 30, 0, 17, 10, 27, 0, 31, 0, 27, 16, 21, 0, 45, 0, 28


Questions:

1. Is this sequence already known?
2. Is E(N) equivalent to a classical divisor-function identity in disguise?
3. Does this type of divisor “mirror asymmetry” around sqrt(N) appear anywhere in analytic number theory (for example in divisor-sum splitting methods or related techniques)?

I’m mainly trying to determine whether this statistic is new or just a reformulation of something standard.

 

[binbots]

One detail that surprised me while computing this:


If N=pqN = pqN=pq is a semiprime with primes p<qp < qp<q, then the statistic gives


E(N) = q − p


So for example:


N = 15 = 3×5
divisors below sqrt(15) are 1 and 3


W(15) = (15/1 − 1) + (15/3 − 3)
= 14 + 2
= 16


so


E(15) = 16 − 14 = 2 = 5 − 3


Similarly:


N = 21 = 3×7 gives E(21) = 4
N = 35 = 5×7 gives E(35) = 2


So the statistic seems to recover the internal prime gap for semiprimes directly from divisor symmetry around sqrt(N).


I’m especially curious whether that behavior appears anywhere in standard divisor-sum techniques.