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