A066376 Number of [*]-divisors d <= n such that there is another [*]-divisor d' < n with d [*] d' = n.

0, 1, 1, 2, 1, 3, 2, 3, 1, 3, 1, 5, 1, 5, 4, 4, 1, 3, 1, 5, 2, 3, 1, 7, 1, 3, 3, 8, 1, 9, 7, 5, 1, 3, 1, 5, 1, 3, 1, 7, 1, 5, 1, 5, 3, 3, 3, 9, 1, 3, 3, 5, 1, 7, 3, 11, 1, 3, 3, 14, 3, 15, 13, 6, 1, 3, 1, 5, 1, 3, 1, 7, 2, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 8, 4, 3, 1, 7, 1, 7, 3, 5, 1, 7, 5, 11, 1

Offset: 1

Author

Marc LeBrun, Dec 22 2001

Keywords

• nonn
• easy
• nice

Comments

Define [+] to be binary bitwise inclusive-OR and let [*] denote the shift-and-[+] product. ([+] is usually simply called OR.) Note that [*] is commutative, associative, and distributes over [+]. If x [*] y = z, we say x and y are [*]-divisors of z.

Examples

			14 has 5 [*]-divisors: 1, 2, 3, 6, 7, since for example 2 [*] 7 = 10 [*] 111 = 1110 OR 0000 = 1110; and 3 [*] 6 = 11 [*] 110 = 1100 OR 0110 = 1110.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A067139 ("primes").

See A003986 for a table of [+] sums, A067138 for a table of [*] products.

Programs

Haskell

import Data.Bits (Bits, (.|.), shiftL, shiftR)
a066376 :: Int -> Int
a066376 n = length [d | d <- [1..n-1], any ((== n) . (orm d)) [1..n]] where
   orm 1 v = v
   orm u v = orm (shiftR u 1) (shiftL v 1) .|. if odd u then v else 0
-- Reinhard Zumkeller, Mar 01 2013

Extensions

Edited by N. J. A. Sloane, Dec 13 2021

Name corrected by Sean A. Irvine, Oct 10 2023