Number-Divisors Almost = ln(m) + 2c-1

[Hans Havermann]

I wrote:

> 1, 2, 3, 5, 7, 11, 17, 19, 23, 47, 89, 125, 131, 203, 219, 455, 1475, 
> 2867, 4649, 7291, 36893, 378878, 517914, 693028, 923373, 1835331, 
> 3147909, 3356513, 3506524, 6782094, 20454813, 25494256, 27802807, 
> 28081980, 47214722, 176344865, 225147772, ...
>
> The value at the last point is 7.105427357601 * 10^-15.

On Dec 4, 2003, at 6:10 PM, Leroy Quet wrote:

> ...but using Length[Divisors] would be better in this case, where
> precision is essential, if Mathematica evaluated
> sum-of-powers-of-divisors numerically.

One of the nice things about Mathematica is that it will not 
approximate unless you ask it to. I *did* ask it to evaluate the 
function at 225147772 and, without giving it too much thought, passed 
on that result (above). Unfortunately, that value is not correct.

The value at 225147772 is (exactly) 2295009911 / 112573886 -2 
EulerGamma - Log[225147772].

When you ask Mathematica to evaluate a (symbolic) number, you generally 
just ask for N[number]. Alternatively, N[number, p] attempts to give a 
result with p-digit precision... 7.105427357601 * 10^-15 is the former 
"approximation" of our result at 225147772 and, as it turns out, not 
really something I should have passed on (implying all that accuracy). 
Asking for 50-digit precision shows that the value at 225147772 is 
actually 1.0619581064343867502657729848045688951270405275786 * 10^-14. 
There's a lot of hair-pulling by amateurs like myself about why 
N[number] does what it does, but that's the way it is.

Fortunately, I looked for new "record" lows at a strictly symbolic 
level, so the sequence stands.