Don Reble has found a proof of the identity. (I want to check it carefully, and will then add it to one of the relevant sequences in the OEIS.) On Thu, Nov 19, 2020 at 10:54 PM Neil Sloane <njasloane@gmail.com> wrote:
Dear Math Fun, The OEIS is full of assertions whose status is unclear - are they theorems or conjectures? This one is stated unconditionally but without a proof. If we had a proof it would help with quite a lot of other sequences. It is surely true, and maybe not difficult to prove.
Can someone help? T(k) = k*(k+1)/2, k >= 0, is a triangular number.
Claim: For n >= 1, we have
Sum_{ k >= 1, stop when T(k-1) >= n } (-1)^(k+1) * ( floor ( (n - T(k-1))/k ) )^2
= Sum_{k=1..n} sigma(k).
The RHS is the sum of the divisors of all numbers from 1 to n, A024916(n), which can also be written as Sum_{d=1..n} d*floor(n/d) = n^2 - Sum_{d=1..n} n%d.
The summands on the LHS look like [ (n-T(k-1))/k ]^2, and the sum stops when the quantity n - T(k-1) becomes zero or goes negative.
For example, when n=8, the assertion is that 8^2 - [ (8-1)/2 ]^2 + [ (8-3)/3 ]^2 = 8^2 - 3^2 + 1^2 = 56 = 1+3+4+7+6+12+8+15 .
Neil