an integer-analog of ln(m!)

[Leroy Quet]

Let a(m) = sum{k=1 to m} floor(ln(k)),

an integer-analog of ln(m!).

a(m) also is, for m = positive integer,

m*floor(ln(m)) - sum{k=1 to floor(ln(m))} floor(e^k).


Some questions:

First, is {a(k)} in the EIS?

Second, what is

ln(m!) - a(m) = sum{k=1 to m} {ln(k)}

asymptotical to?

({x} is the fractional part of x.)

Or, I might ask instead, what is

limit{m-> oo}  (ln(m!) - a(m))/m  ?


thanks,
Leroy
Quet