[math-fun] An identity involving the floor function
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
Someone asked me to give more examples, so here are the first 10 cases of the identity: 1: 1 = 1 2: 2^2 = 1+3 =4 3: 3^2-1^2 = 1+3+4 = 8 4: 4^2-1^2 = " + 7 = 15 5: 5^2-2^2 = " + 6 = 21 6: 6^2-2^2+1^2 = " + 12 = 33 7: 7^2-3^2+1^2 = " + 8 = 41 8: 8^2-3^2+1^2 = " + 15 = 56 9: 9^2-4^2+2^2 = " + 13 = 69 10: 10^2-4^2+2^2-1^2 = " + 18 = 87 For the LHS (before squaring) see A235791, and the RHS is A024916 (partial sums of sigma(n)) 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
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
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Neil Sloane