A080942 Number of divisors of n that are also suffixes of n in binary representation.

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 4, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2

Offset: 1

Author

Reinhard Zumkeller, Feb 25 2003

Keywords

• nonn
• base

Comments

a(n) = 1 iff n = 2^k (A000079), the only divisor is n itself.

For a(n) > 1 the other trivial divisor is 1 for odd numbers and 2 for even numbers (A057716).

Examples

			n=63 has A000005(63)=6 divisors: 1='1', 3='11', 7='111', 9='1001', 21='10101' and 63='111111', {1,11,111,111111} are also suffixes of 111111, therefore a(63)=4.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007088, A080940, A080941, A080943, A080945, A080946, A080947, A080948.

Cf. A000005, A027750, A030308, A057716, A239826.

Programs

Haskell

import Data.List (isPrefixOf); import Data.Function (on)
a080942 n = length $
            filter ((flip isPrefixOf `on` a030308_row) n) $ a027750_row n
-- Reinhard Zumkeller, Mar 27 2014

Mathematica

a[n_] := DivisorSum[n, 1 &, Mod[n, 2^BitLength[#]] == # &]; Array[a, 100] (* Amiram Eldar, Apr 07 2023 *)

Python

from sympy import divisors
def A080942(n): return sum(1 for d in divisors(n,generator=True) if not (d^n)&((1<Chai Wah Wu, Jun 20 2023

Formula

a(A080943(n)) = 2.

a(A080945(n)) > 2.

a(A080946(n)) = 3.

a(A080947(n)) > 3.

a(n) <= A000005(n).

a(p) = 2 for odd primes p.

a(A080948(n)) = n and a(m) < n for m < A080948(n).