How to find a maximal odd decomposition of integer M?

Question

Let M be an integer in range [1; 1,000,000,000].

A decomposition of M is a set of unique integers whose sum is equal to M.

A decomposition is odd

Answer 1

Here is an idea which should work. It requires a net removal of at most 1 number from the greedy set you construct.

Construct your O(sqrt(M)) list as usual (without result[len - 1] = M - sum;). If the sum is not a square number (i.e. exact):

• Add the next largest odd number

• Take the difference between your sum now and your target number -> N

  • If N is odd, remove the corresponding number
  • If N is even but not 2, remove the largest odd number smaller than N, and the 1
  • If N is 2, remove the penultimate number you just added. This will make the difference 0

Proof:

• If you add the consecutive odd numbers sequentially then the list should be as dense as possible - i.e. you cannot have a larger list
• Also the difference is always bounded by the odd number you add in step 2, so a net removal of up to two numbers is always accounts of either odd or even differences.

Examples:

• Say you want 42: construct [1, 3, 5, 7, 9, 11], add 13 => sum = 49, remove 7 (no net removal)
• You want 39: construct [1, 3, 5, 7, 9, 11], add 13, remove 1 and 9 (net removal of 1)
• You want 38: construct [1, 3, 5, 7, 9, 11], add 13, remove 11 (no net removals)

EDIT: error in last case corrected courtesy of @user1952500