A066241 1 + number of anti-divisors of n.

1, 1, 2, 2, 3, 2, 4, 3, 3, 4, 4, 3, 5, 4, 4, 3, 6, 5, 4, 4, 4, 6, 6, 3, 6, 4, 6, 6, 4, 4, 6, 7, 6, 4, 6, 3, 6, 8, 6, 5, 5, 6, 6, 4, 8, 6, 6, 4, 7, 7, 4, 8, 8, 4, 6, 4, 6, 8, 8, 7, 5, 6, 8, 3, 6, 6, 10, 8, 4, 6, 6, 7, 8, 6, 6, 6, 10, 6, 4, 6, 7, 8, 8, 5, 9, 6, 8, 8, 4, 6, 6, 6, 8, 10, 10, 2, 8, 9, 6, 5

Offset: 1

Author

N. J. A. Sloane, Dec 31 2001

Keywords

• nonn
• easy

Comments

See A066272 for definition of anti-divisor.

Examples

			For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd divisors > 1 {3,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the anti-divisors of 12 are 4, 5, 7, 12. Therefore a(18) = 1 + 4 = 5.
		

Links

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

Crossrefs

Cf. A058838. Equals 1 + A066272(n).

Programs

Mathematica

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 &]; Table[ Length[ antid[n]] + 1, {n, 1, 100} ]

Extensions

More terms from Robert G. Wilson v, Jan 03 2002