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

[Hans Havermann]

On Dec 3, 2003, at 6:02 AM, Pfoertner, Hugo wrote:

>  n  delta
>  1  0.8455687
>  2  0.652421519
>  3  0.413623078
>  5  0.236130788
>  7  0.185372837
> 11  0.084037064
> 17  0.071178885
> 19  0.059024458
> 23  0.01442231
> 47  -0.004578902
> 89  -0.002618232

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.

Earlier, I opined:

> Mathematica's DivisorSigma does sum of (powers of) divisors. For 
> number-of-divisors, I use Length[Divisors].

D'oh. The sum of the *zero* powers of the divisors *is* the number of 
divisors. Turns out this is actually a bit faster than 
Length[Divisors].