1,1

Additional conjectures suggested by computational experiments: 1) Numbers all of whose anti-divisors (ADs) are odd => {2^k} (A000079). 2) Numbers with AD 2, all other ADs odd => primes (A000040). 3) Numbers none of whose ADs are multiples of 3 => 3*2^k (A007283). 4) Numbers all of whose ADs are even => 3*A002822 = A040040 (except for a(0)=1), both related to twin prime pairs.

Calculations suggest the following conjecture. This sequence consists of all odd primes and nonnegative powers of 2 and no other terms. This has been verified for to n=100000. Robert G. Wilson v extended the conjecture out to 2^20.

See A066272 for definition of anti-divisor.

Jon Perry, The Anti-Divisor

Jon Perry, The Anti-divisor [Cached copy]

Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]

Calculation suggest the following conjecture: The sequence consists of all primes and nonnegative powers of 2 and no other terms. This has been verified for to n=100000.

antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n - 1], OddQ[ # ] && # != 1 &], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 &], 2n/Select[ Divisors[2*n], OddQ[ # ] && # != 1 &]]], # < n & ]; f[n_] := Select[ antid[n], EvenQ[ # ] && # > 2 & ]; Select[ Range[3, 300], f[ # ] == {} & ]

Cf. A000040, A000079, A002822, A007283, A040040.

Cf. A014545, A006862.

Sequence in context: A138494 A111801 A108372 this_sequence A003310 A038525 A057201

Adjacent sequences: A066539 A066540 A066541 this_sequence A066543 A066544 A066545

nonn

John W. Layman (layman(AT)math.vt.edu), Jan 07 2002