Search for a number theoretic function related to "prime divisor sums"

[juergen]

Hi all,

does anybody know a number theoretic function similar to the divisor sum, but which only sums up the divisors which are prime?

For me a function would be interesting, which builds the

1. Sum over p/x for all primes p which divide x where p<>x

or

2. Sum over p for all primes p which divide x where p<>x (the same as aove * x)

Of course I could invent this function myself, but I hope that it already exists and maybe there are already some interestings fact known about this function which I could use :o)

I am especially interested in Numbers x which have a Sum (1.) >= 1 in the function mentioned above (or 2. >= x). Does anybody know something about such functions or such numbers?

Thank you in advance

Juergen

[juergen]

Originally Posted by juergen

Hi all,

does anybody know a number theoretic function similar to the divisor sum, but which only sums up the divisors which are prime?

For me a function would be interesting, which builds the

1. Sum over p/x for all primes p which divide x where p<>x

or

2. Sum over p for all primes p which divide x where p<>x (the same as aove * x)

I am especially interested in Numbers x which have a Sum (1.) >= 1 in the function mentioned above (or 2. >= x). Does anybody know something about such functions or such numbers?

Thank you in advance

Juergen

Hi all,

sorry I missed something in the second form. It is wrong :o(
The first form is correct and the condition Sum (1.) >= 1.

I am trying to learn something about numbers x for which this sum is greather than x. One example is

30 <= 31

30=2*3*5
31=30/2 + 30/3 + 30/5 = 15 + 10 + 6

Does anybody know something about such numbers?

Thank you in advance

Juergen

[juergen]

Hi all,

the sum over 1/p for all prime p < x is called the reciprocal prime sum. If you are interested you will find further info on mathworld by searching for "mertens constant".

regards

Juergen