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

[Hans Havermann]

On Dec 2, 2003, at 1:44 PM, <all at abouthugo.de> wrote:

> I checked from 2..128 and found records for
>  3: 2 -  ln(3) +1 -2*0.5772156649=0.746956382
>  5: 2 -  ln(5) +1 -2*0.5772156649=0.236130758
>  7: 2 -  ln(7) +1 -2*0.5772156649=-0.100341479
> 46: 4 - ln(46) +1 -2*0.5772156649=0.01692727

I get: 1, 3, 5, 7, 46, 2514, 2522, 2526, 2534, 2536, 2542, 2546, 2553, 
2555, 18873, 139454, 139475, 7614005, 7614010, 7614015, 7614022, 
7614030, 7614033, 7614034, 7614056, 7614062, 7614066, 7614069, 7614079, 
7614082, 7614086, 7614087, 7614088, 7614104, 7614113, 7614114, 7614123, 
7614127, 7614130, 7614135, 7614136, 7614139, 7614142, 7614145, 7614158, 
7614165, 7614166, 7614168, 7614174, 7614184, 7614190, 7614195, 7614219, 
7614232, 7614245, 7614249, 7614255, 7614257, 7614265, 7614266, 7614269, 
7614280, 7614298, 7614310, 7614312, 7614318, 7614321, 7614326, 7614327, 
7614330, 7614344, 7614345, 7614357, 7614365, 7614385, 7614386, 7614392, 
7614395, 7614397, 7614398, 7614406, 7614408, 7614411, 7614415, 7614422, 
7614435, 7614456, 7614459, 7614474, 7614481, 7614483, 7614489, 7614490, 
7614498, 7614526, ...

For the last one, the value is 3.733908 * 10^-7.

> Should be very easy to extend this with Maple's tau,
> Mathematica's DivisorSigma or Pari's numdiv ...?

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