
COMMENTS

See A066272 for definition of antidivisor.
Jon Perry calls these antiprimes.
Except for a(2) = 4, the terms of A066466 have form 2^k*p where p is odd prime and both 2^(k+1)*p1, 2^(k+1)*p+1 are prime (i.e., twin primes). In other words, this sequence, omitting 4, is a subsequence of A040040 containing elements of the form 2^k*p with prime p.
Furthermore, since 2^(k+1)*p1, 2^(k+1)*p+1 must equal 1 and +1 modulo 3, the number 2^(k+1)*p must be 0 modulo 3, implying that p=3. Therefore every term, except 4, must be of the form 3*2^k such that 3*2^(k+1)1, 3*2^(k+1)+1 are twin primes.
According to these tables: http://www.prothsearch.com/riesel1.html and http://www.prothsearch.com/riesel2.html there are no other such k up to 18*10^6. Therefore a(6) (if it exists) is greater than 3*2^(18*10^6) ~= 10^5418540. (End)
The 2 last known antiprimes seem to relate to the Fermat primes (coincidence?):
96 = 3 * 2^5 = 3 * 2^F_1 = 3 * 2^[2^(2^1) + 1] and
393216 = 3 * 2^17 = 3 * 2^F_2 = 3 * 2^[2^(2^2) + 1],
where F_k is the kth Fermat prime. (End)


MATHEMATICA

antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n  1], OddQ[ # ] && # != 1 & ], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 & ], 2n/Select[ Divisors[ 2n], OddQ[ # ] && # != 1 &]]], # < n & ]]; Select[ Range[10^5], Length[ antid[ # ]] == 1 & ]
