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

1,3

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).

Links

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

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).

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

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

Sequence in context: A043529 A201219 A254315 * A099812 A246600 A068068

Adjacent sequences: A080939 A080940 A080941 * A080943 A080944 A080945

Keyword

nonn,base

Author

Reinhard Zumkeller, Feb 25 2003

Status

approved