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%I #19 May 23 2023 23:28:46
%S 3,4,6,96,393216
%N Numbers having just one anti-divisor.
%C See A066272 for definition of anti-divisor.
%C Jon Perry calls these anti-primes.
%C From _Max Alekseyev_, Jul 23 2007; updated May 23 2023: (Start)
%C Except for a(2) = 4, the terms of A066466 have form 2^k*p where p is odd prime and both 2^(k+1)*p-1, 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.
%C Furthermore, since 2^(k+1)*p-1, 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.
%C 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)
%C From _Daniel Forgues_, Nov 23 2009: (Start)
%C The 2 last known anti-primes seem to relate to the Fermat primes (coincidence?):
%C 96 = 3 * 2^5 = 3 * 2^F_1 = 3 * 2^[2^(2^1) + 1] and
%C 393216 = 3 * 2^17 = 3 * 2^F_2 = 3 * 2^[2^(2^2) + 1],
%C where F_k is the k-th Fermat prime. (End)
%H Jon Perry, <a href="/A066272/a066272a.html">The Anti-divisor</a> [Cached copy]
%H Jon Perry, <a href="/A066272/a066272.html">The Anti-divisor: Even More Anti-Divisors</a> [Cached copy]
%t 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 & ]
%Y Cf. A066272.
%Y A066272(a(n)) = 1.
%Y Cf. A000215, A019434. - _Daniel Forgues_, Nov 23 2009
%K nonn,hard
%O 1,1
%A _Robert G. Wilson v_, Jan 02 2002
%E Edited by _Max Alekseyev_, Oct 13 2009
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