Finding if an integer is odd through Riemann or some function?

caljuice

I'm trying to find if a number is odd or not, basically if X % 2 = 1.

Can this be expressed through some function? Like the sum of 1 + 2 .. +n is n(n+1) /2

Or as a Riemann sum?

I'm trying to add only the odd numbers from a random set of N integers to a sum.

SteveL27

Do you mean in some programming language? If '/' is integer division, then x is even if and only if

x = 2 * (x/2). [using '=' as equality, not assignment]

But if you already have a mod operator like % then you can just use that.

phillip1882

i'm not entirely sure what you're getting at either, but
2*n+1 is always an odd number, for any value of n.

phillip1882

ohh okay i think i see what you're saying, you have a set of random numbers, and you only want to add the odd ones.
yes the modulus function works well.
if(x%2 == 1){
...
}
alternatively, you can test the last bit to see if it's 1.
if(x&1 == 1){
...
}
this is slightly faster.

caljuice

Not what I meant but I did word this problem poorly. It's actually probably a bad and weird question anyway. I appreciate the help though

For example
Let's say the set of integers is from some function( not random anymore)
[itex]\sum \limits_{i=1}^n A[i] (A[i]\mod{2})[/itex] is only adding odd terms but the equation is not reducible, so I was hoping the % could be expressed differently so I could solve or reduce the problem. If finding the number was odd or mod or % 2 was rewritable as a function or riemann sum, then I could reduce or solve the problem.

pwsnafu

I fail to see how Riemann sums would be relevant in what appears to be a discrete math question.

coolul007

I think he wants do something like the trick with minus signs alternating by using n for an exponent on -1. Just a guess here.

pwsnafu

So something like
##\frac{1 - (-1)^n}{2}##

coolul007

adding all the odds would look like this:

[itex]\sum n(\frac{1 - (-1)^n}{2})[/itex]

where n is a positive integer

caljuice

Ya that works, thanks guys. Sorry about the poor wording. I used that equation daily in my fourier series class, I'm surprised I forgot about that after one summer.

ramsey2879

I don't understand the problem here. All odd numbers equal 1 mod 2, so adding n odd numbers will give you the result in mod 2 the following:

[itex]\sum \limits_{i=1}^n A[i]\mod{2}) = n [/itex] so n odd numbers will give an even number if n is even and an odd number if n is odd. There is no need to use more complex functions.

pwsnafu

He doesn't want to count the number of odd A[i], but to sum only the odds. So if
##A = (1,2,3,5,7,8,10)##, he wants an expression which evaluates 1+3+5+7.

SteveL27

Oh I see. Well then how about

x = [itex]\displaystyle \sum_{k=1}^n a_ksin^2(a_k\pi) [/itex]

If a_k is even, sin(a_k[itex]\pi[/itex]) is 0; and if it's odd, sin(a_k[itex]\pi[/itex]) is +/- 1.

Eval

Hmm, this looks neat, so you are going to add all the odd terms in f(x)? So you need, as a function, [itex]\displaystyle \sum_{x=a}^n f(x)\frac{1-(-1)^{f(x)}}{2}[/itex]. That looks like a really cool and useful function! I have literally filled notebooks finding sums like that, so I have a huge collection, but I've never used a power of f(x) for anything other than f(x)=c or f(x)=x. Do you mind if I play with this? The most that I can simplify this too, without any extra exploration is:
[itex]\displaystyle \sum_{x=a}^n f(x)\frac{1-(-1)^{f(x)}}{2} = \sum_{x=a}^n \frac{f(x)}{2}-\sum_{x=a}^n f(x)\frac{(-1)^{f(x)}}{2}[/itex]


For my own purposes, I will want to look at x=0, specifically. If I want to look at specific parts of (f(x), I will use f(x-c).

EDIT: I realised after working through this a bit that I forgot to multiply by f(x). The formulas above are corrected (I hope). However, I do have a list of solutions that relate to the correct version (where f(x)=x^a for some non-negative integer a). Typically, the first part of that sum would be calculated with the Hurwitz Zeta function and the Riemann Zeta function (though I have a different way that I prefer). The second part, I believe, can be similarly computed. For cases where f(x) is a polynomial, this should not be difficult to compute, especially as it can be expressed as a linear combination of c_nxⁿ