A066466 Numbers having just one anti-divisor.

3, 4, 6, 96, 393216

Offset: 1

Author

Robert G. Wilson v, Jan 02 2002

Keywords

• nonn
• hard

Comments

See A066272 for definition of anti-divisor.

Jon Perry calls these anti-primes.

A066272(a(n)) = 1.

From Max Alekseyev, Jul 23 2007; updated Jun 25 2025: (Start)

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.

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. In other words, k+1 belongs to the intersection of A002253 and A002235.

According to Ballinger and Keller's lists, there are no other such k up to 22*10^6. Therefore a(6) (if it exists) is greater than 3*2^(22*10^6) ~= 10^6622660. (End)

From Daniel Forgues, Nov 23 2009: (Start)

The 2 last known anti-primes 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 k-th Fermat prime. (End)

Links

• Ray Ballinger and Wilfrid Keller, List of primes k·2^n + 1 for k < 300
• Wilfrid Keller, List of primes k·2^n - 1 for k < 300
• Jon Perry, The Anti-divisor [Cached copy]
• Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]

Crossrefs

Cf. A000215, A002253, A002235, A066272, A019434.

Programs

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 & ]

Extensions

Edited by Max Alekseyev, Oct 13 2009