1. ## Just Research...........

I am doing a research relatively with anti-divisors and the “opposite” of them numbers .
Actually I am trying to count (to find a formula which counts) the number of pairs*of type (“anti divisor odd composite number less than half of an integer a”---“odd composite number greater than the half of an integer a”) such as the addition of them has as result the integer a.

For example :24=9+15
9<24/2 is odd comp. and does not divide 24
and 24>15>24/2 is and odd comp.
So there is one pair * of integers of the this specific type for the number 24...
I have done some work on that but I didn’t yet found one way to count exactly the number of ways which an integer can be written on this way.
the only thing I know about that is a lower bound for this number but its not really useful.I t would be more useful to know the upper bound value
Does anyone knows anything about that?
Thanks!!

2. Originally Posted by Apostolos
I am doing a research relatively with anti-divisors and the “opposite” of them numbers .
Actually I am trying to count (to find a formula which counts) the number of pairs*of type (“anti divisor odd composite number less than half of an integer a”---“odd composite number greater than the half of an integer a”) such as the addition of them has as result the integer a.

For example :24=9+15
9<24/2 is odd comp. and does not divide 24
and 24>15>24/2 is and odd comp.
So there is one pair * of integers of the this specific type for the number 24...
I have done some work on that but I didn’t yet found one way to count exactly the number of ways which an integer can be written on this way.
the only thing I know about that is a lower bound for this number but its not really useful.I t would be more useful to know the upper bound value
Does anyone knows anything about that?
Thanks!!
Let $\displaystyle O_{1/2}(n)$ be the be one less than the number of odds less than $\displaystyle n/2$ (so we are not counting $\displaystyle 1$ as it is a unit and we are going to be interested in composits and primes only). and let $\displaystyle p_c(x)$ be a prime counting function that gives the number of primes less than $\displaystyle x$.

Also let $\displaystyle AD(n)$ denote the number of pairs $\displaystyle (a,b)$ with $\displaystyle a$ and $\displaystyle b$ composite, $\displaystyle a<n/2$ and $\displaystyle n=a+b$. Then:

$\displaystyle AD(n) \le O_{1/2}(n)- p_c(n/2)$

If I have done this right, the right hand side of this inequality should be the number of odd composites less than $\displaystyle n/2$.

RonL

3. Originally Posted by Apostolos
I am doing a research relatively with anti-divisors and the “opposite” of them numbers .
Actually I am trying to count (to find a formula which counts) the number of pairs*of type (“anti divisor odd composite number less than half of an integer a”---“odd composite number greater than the half of an integer a”) such as the addition of them has as result the integer a.

For example :24=9+15
9<24/2 is odd comp. and does not divide 24
and 24>15>24/2 is and odd comp.
So there is one pair * of integers of the this specific type for the number 24...
I have done some work on that but I didn’t yet found one way to count exactly the number of ways which an integer can be written on this way.
the only thing I know about that is a lower bound for this number but its not really useful.I t would be more useful to know the upper bound value
Does anyone knows anything about that?
Thanks!!
the problem is very difficult! an upper bound that can be proved quite easily is this:

$\displaystyle f(n) \leq \min \{[n/4] + \omega_1(n) + 1 - \tau_1(n) -\pi (n/2 - 1), [n/4]+ \pi (n/2) - \pi (n) \},$

where $\displaystyle f$ is the fuction you're looking for, $\displaystyle [ \ \ ]$ is the floor function, $\displaystyle \omega_1(n)$ is the number

of (distinct) odd prime factors of n, $\displaystyle \tau_1(n)$ is the number of odd divisors of n, and finally

$\displaystyle \pi(x)$ is the prime counting function.
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