This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A066466 #28 Jun 27 2025 01:10:18 %S A066466 3,4,6,96,393216 %N A066466 Numbers having just one anti-divisor. %C A066466 See A066272 for definition of anti-divisor. %C A066466 Jon Perry calls these anti-primes. %C A066466 A066272(a(n)) = 1. %C A066466 From _Max Alekseyev_, Jul 23 2007; updated Jun 25 2025: (Start) %C A066466 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 A066466 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. %C A066466 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) %C A066466 From _Daniel Forgues_, Nov 23 2009: (Start) %C A066466 The 2 last known anti-primes seem to relate to the Fermat primes (coincidence?): %C A066466 96 = 3 * 2^5 = 3 * 2^F_1 = 3 * 2^[2^(2^1) + 1] and %C A066466 393216 = 3 * 2^17 = 3 * 2^F_2 = 3 * 2^[2^(2^2) + 1], %C A066466 where F_k is the k-th Fermat prime. (End) %H A066466 Ray Ballinger and Wilfrid Keller, <a href="http://www.prothsearch.com/riesel1.html">List of primes k·2^n + 1 for k < 300</a> %H A066466 Wilfrid Keller, <a href="http://www.prothsearch.com/riesel2.html">List of primes k·2^n - 1 for k < 300</a> %H A066466 Jon Perry, <a href="/A066272/a066272a.html">The Anti-divisor</a> [Cached copy] %H A066466 Jon Perry, <a href="/A066272/a066272.html">The Anti-divisor: Even More Anti-Divisors</a> [Cached copy] %t A066466 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 A066466 Cf. A000215, A002253, A002235, A066272, A019434. %K A066466 nonn,hard %O A066466 1,1 %A A066466 _Robert G. Wilson v_, Jan 02 2002 %E A066466 Edited by _Max Alekseyev_, Oct 13 2009