Challenges

[bplus]

since we are a page beyond the original challenge post here it is again:

What numbers between 1 and 1000 can NOT be expressed as the sum of consecutive integers?

ie 15 = 7 + 8
18 = 5 + 6 + 7

so neither one of these are in the set of numbers that can NOT be expressed as a sum of consecutive integers.

BONUS: one number can be expressed 15 different ways as a sum of consecutive numbers what is it and list the ways. Smile

here are some hints i posted elsewhere in case someone else would like to give this a shot.

hint: 1 + 2 + 3... + n = n(n+1)/2 so sum of first 100 pos integers = 100 * 101 / 2 = 50 * 101 = 5050

my first approach was to eliminate all 2 consecutive sums first then 3, then 4... up to about 40 like sieving for primes, that worked.
then i worked out a way to list all ways to get consecutive numbers to add up to any given number, that too was successful in 15 lines of code.

n + (n + 1) = sum of consecutive numbers
n + (n + 1) + (n + 2) = sum of 3 consecutive numbers
n + (n + 1) + (n + 2) + (n + 3) sum of 4 consecutive numbers

2nd day reply at another forum

yeah there are 10 from 1 to 1000 inclusive, funny i did not count them before i reread you first reply.

did you find the one that has 15 different ways to sum consecutives to it?
yes the sum of digits of the special number is 18
i didn't read that correctly the first time sorry.
In fact, what you wrote made no sense to me, but I see and appreciate you are keeping things semi-secret

more hints;
1+2 = 3
2+3 = 5
3+4 = 7
...

so all 2 consecutive number sums are odd.

1+2+3 = 6
2+3+4 = 9
3+4+5 = 12
...

so every other 3 consecutive number sums are even = 3n, n starts at 2

if 2 consecutive sums are even so likely then 4 consecutive sums
1+2+3+4 = 10
2+3+4+5 = 14
3+4+5+6 = 18
...

yeah 2 consecutives already took out all these numbers

so what's interesting to us is sum of odd number consecutive integers

last hint:

i have been thinking the pattern pretty much makes itself known in first 10 integers, by 20 it is pretty clear and by 100 solid!
but we need something for our computer programming skills to be sharpened so the BONUS number question.

i will post my code sunday, but i wonder if someone will try an approach similar to mine?