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A066466 Numbers having just one anti-divisor. 5

%I

%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: (Start)

%C Except for the term 4, the elements 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 element 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://web.archive.org/web/20161028080239/http://www.prothsearch.net/riesel.html and http://web.archive.org/web/20161028021640/http://www.prothsearch.net/riesel2.html there are no other such k up to 5*10^6. Therefore the next element of A066466 (if it exists) is greater than 3*2^(5*10^6) ~= 10^1505150. (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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Last modified July 13 03:51 EDT 2020. Contains 335673 sequences. (Running on oeis4.)