Problem 643

[vapor]

What does g(100)=1031 mean in this problem?
Update: It seems that setting p<q gives the correct result.

[mscha]

I get f(10²) = 1037, not 1031, given the definition in the problem.
If I change the definition to require 1 ≤ p < q ≤ n instead of 1 ≤ p ≤ q ≤ n, then I get 1031. (That excludes (2,2), (4,4), (8,8) ... (64,64).)
So which is wrong, the definition or the example?

[MHealy]

So which is wrong, the definition or the example?

The definition is wrong; the example values are correct. (Based on the expected answer.)

[hk]

It should read $1\le p\lt q\le n$. Problem edited.

[vamsikal3]

<deleted post>

[RobertStanforth]

The correct text is: For example, 24 and 40 are 2-friendly because ....

Thanks, Vamsi. This is fixed now.

[Cot-O-Bus]

Looks like expansion of PI^-2, but...

[HashFunction]

How come $$gcd(a,b)=2^t,t>0$$ , but with brute force I get f(100) = 1031 when t = 0 , that is $$gcd(a,b) = 2^0 = 1$$ , should t be >= 0 or am I wrong ?

[hk]

Could it be that your bruteforcer is wrong?

[Derfellios]

... because $\gcd(24,36) = 12 = 2^2\cdot 3$ not a power of $2$.

This part of the second sentence seems grammatically incorrect to me (a non-native English speaker). I would write

... because $\gcd(24,36) = 12 = 2^2\cdot 3$ is not a power of $2$.

[RobertStanforth]

... because $\gcd(24,36) = 12 = 2^2\cdot 3$ not a power of $2$.

This part of the second sentence seems grammatically incorrect to me (a non-native English speaker). I would write

... because $\gcd(24,36) = 12 = 2^2\cdot 3$ is not a power of $2$.

Thank you for pointing this out. I have corrected it, inserting the word "is" as you suggested.