🚀 go-pugleaf

Date: Wed, 5 Aug 87 11:53:41 edt
From: mtu!siskowit!swgraham@siskowit.uucp (Sidney Graham)
Subject: A problem on sums of divisors
     
   Here is a question posed to me by Don Saari, who is an applied mathematician
at Northwestern. I would appreciate any solutions, hints, or references
to the literature.
     
   Let  f(n)  be the sum of all the aliquot divisors of  n , i.e. the sum
of all the divisors of  n  except  n  itself. The function  f  maps
the positive integers into the positive integers. The question is: What
is the range of  f?
     
    That  f is not onto follows from the observation that  2  is not in
the range of  f. I can show that almost all odd integers are in the range
of  f. Let  N  be an odd integer.  If
     
(1)     N-1 is the sum of two distinct odd primes p and q,
     
then  f(pq) = N . Statement (1) is a slight variant of the Goldbach conjecture,
and it is known to be true for almost all odd  N  (see Chapter 3.2 of
^The Hardy-Littlewood Method ^ by Bob Vaughan). (1) is probably true for
all odd N > 7 . I can't say much about what even integers are in the
range of  f . I conjecture that at least a positive density of the even
integers are not in the range.  Actually, "conjecture" is too nice a word;
it's more like a wild guess.