Definitions

   Untouchable numbers   are also known as   nonaliquot numbers.
   An   untouchable number   is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer.   (From Wikipedia)
   The   sum of all the proper divisors   is also known as   the   aliquot sum.
   An   untouchable   are those numbers that are not in the image of the aliquot sum function.   (From Wikipedia)
   Untouchable numbers:   impossible values for the sum of all aliquot parts function.   (From OEIS:   The On-line Encyclopedia of Integer Sequences®)
   An untouchable number is a positive integer that is not the sum of the proper divisors of any number.   (From MathWorld™)


Observations and conjectures

All untouchable numbers   >  5  are composite numbers.

No untouchable number is perfect.

No untouchable number is sociable.

No untouchable number is a Mersenne prime.

No untouchable number is   one more   than a prime number,   since if   p   is prime,   then
the sum of the proper divisors of   p2   is  p + 1.

No untouchable number is   three more   than an odd prime number,   since if   p   is an odd prime,   then the
sum of the proper divisors of   2p   is  p + 3.

The number  5  is believed to be the only odd untouchable number,   but this has not been proven:   it would follow from a
slightly stronger version of the   Goldbach's conjecture,   since the sum of the
proper divisors of   pq   (with   p, q   being
distinct primes)   is   1 + p + q.

There are infinitely many untouchable numbers,   a fact that was proven
by   Paul Erdős.

According to Chen & Zhao,   their natural density is at least   d > 0.06.



Task

   show  (in a grid format)  all untouchable numbers  ≤  2,000.
   show (for the above)   the   count   of untouchable numbers.
   show the   count   of untouchable numbers from unity up to   (inclusive):
                   10
                 100
               1,000
             10,000
           100,000
   ... or as high as is you think is practical.
   all output is to be shown here, on this page.


See also

   Wolfram MathWorld:   untouchable number.
   OEIS:   A005114 untouchable numbers.
   OEIS:   a list of all untouchable numbers below 100,000   (inclusive).
   Wikipedia: untouchable number.
   Wikipedia: Goldbach's conjecture.


f = DivisorSigma[1, #] - # &;
limit = 10^5;
c = Not /@ PrimeQ[Range[limit]];
slimit = 15 limit;
s = ConstantArray[False, slimit + 1];
untouchable = {2, 5};
Do[
 val = f[i];
 If[val <= slimit,
  s[[val]] = True
  ]
 ,
 {i, 6, slimit}
]
Do[
 If[! s[[n]],
  If[c[[n - 1]],
   If[c[[n - 3]],
    AppendTo[untouchable, n]
    ]
   ]
  ]
 ,
 {n, 6, limit, 2}
]
Multicolumn[Select[untouchable, LessEqualThan[2000]]]
Count[untouchable, _?(LessEqualThan[2000])]
Count[untouchable, _?(LessEqualThan[10])]
Count[untouchable, _?(LessEqualThan[100])]
Count[untouchable, _?(LessEqualThan[1000])]
Count[untouchable, _?(LessEqualThan[10000])]
Count[untouchable, _?(LessEqualThan[100000])]