Can anyone identify these constants?

[Paul D. Hanna]

Seqfans, 
       My guess of the limit_{n-->oo} a(n)/10^n for Eric's sequences,
"Number of decimal digits in the numerator of HarmonicNumber[10^n]"
> http://www.research.att.com/projects/OEIS?Anum=A114467
and "Number of decimal digits in the denominator of HarmonicNumber[10^n]"
> http://www.research.att.com/projects/OEIS?Anum=A114468
> 0.434286...
 
is: Log[10, exp(1) ] = 0.434294481903251827651128918...
 
Can someone prove this? 
 
Thanks,
      Paul
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