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Abstract
Let
σ ( n ) be the sum of
the divisors of
n .
Although much attention has been paid to the possible values of
σ ( n )
−
n (the sum
of proper divisors), comparatively little work has been done on the possible values of
e ( n )
: =
σ ( n )
− 2 n . Here we
present some theoretical and computational results on these values. In particular, we
exhibit some infinite and possibly infinite families of integers that appear in the image of
e ( n ) . We also find
computationally all values of
n
< 1 0 2 0
for which
e ( n )
is odd, and we present some data from our computations. At the end of
this paper, we present some conjectures suggested by our computational
work.
Keywords
sigma function, sum of divisors, excedents, computational
mathematics
Mathematical Subject Classification 2010
Primary: 11A25, 11Y70
Milestones
Received: 7 December 2012
Revised: 18 May 2013
Accepted: 20 May 2013
Published: 8 October 2013
Communicated by Kenneth S. Berenhaut