## Consecutive Sums

### February 12, 2019

Given the positive integer 15, there are three ways to take consecutive positive integers that sum to 15: 1+2+3+4+5, 4+5+6 and 7+8.

Your task is to write a program that, given a positive integer, finds all the ways that consecutive positive integers can sum to the target integer. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

Quite a nifty one, reminiscent of Euler… Here is my take on it, using Julia 1.0:

function SumOfConsecutiveNumbers(x::Int64)

n = div(x,2)

n1 = n + 1

S = Array{Array{Int64}}(undef, n)

c = 0

end

Cheers!

There is another way to solve the problem:

Let the series be k, k+1, k+2, …k+n = N

Then k(n+1) + n(n+1)/2 = N (1)

or (n+1)(n+2k) = 2N (2)

Thus we find all the divisors of 2N and solve for n and k. Some solutions do not give an integral value for k are therefore not valid.

@Ernie: nice solution. Given 2N = ab, k is integral when a and b are of different parities, so we have a (non-trivial) series for N for each odd divisor of N (and this explains Praxis’s observation on 10^n,whose odd divisors are 5^i, 1 <= i <= n).

Here’s some actual code for that solution:

Interval is [k..k+n] for n > 0, sum is k*(n+1)+ n(n+1)/2 = (n+1)(n+2k)/2

so 2N = ab with parity of a and b different. The odd one of a and b must

factor N, and any odd divisor of N will result in an interval sum:

An equivalent function to test cosums – the sum of integers between n and m (inclusive, with n <= m) is (m-n+1)(m+n)/2 decreasing n increases this product and decreasing m decreases the product:

And for a test, print out the numbers where there are more solutions than any lower number:

Looks like we are generating https://oeis.org/A053624, Highly Composite Odd Numbers:

A quick and dirty Haskell version.

Hello Everyone,

This is my first post. I am very new to programming and wanted to improve with some problem solving. Here is my solution using Javascript. I definitely did not make the output pretty. Hope this post works. Thanks

Can’t seem to post my entry

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What in the name of sanity happened to my lovely indentation?

https://en.support.wordpress.com/code/posting-source-code/

ie. do something like this:

@richard: I think I got the formatting fixed properly. Let me know if I did something wrong. The easiest way to post code is just to wrap it with pre … /pre tags.

Here’s a solution in C.

Example Usage: