A226590 Total number of 0's in binary expansion of all divisors of n.

0, 1, 0, 3, 1, 2, 0, 6, 2, 4, 1, 6, 1, 2, 1, 10, 3, 7, 2, 9, 2, 4, 1, 12, 3, 4, 3, 6, 1, 6, 0, 15, 5, 8, 4, 15, 3, 6, 3, 16, 3, 8, 2, 9, 5, 4, 1, 20, 3, 9, 5, 9, 2, 10, 3, 12, 4, 4, 1, 15, 1, 2, 4, 21, 7, 14, 4, 15, 5, 12, 3, 26, 4, 8, 6, 12, 4, 10, 2, 25, 7

Offset: 1

Author

Jaroslav Krizek, Aug 31 2013

Keywords

• base
• nonn

Comments

Also total number of 0's in binary expansion of concatenation of the binary numbers that are the divisors of n written in base 2 (A182621).

a(n) = 0 iff n = 1 or n is a Mersenne prime (A000668). - Bernard Schott, Apr 22 2022

Examples

			a(8) = 6 because the divisors of 8 are [1, 2, 4, 8] and in binary: 1, 10, 100, 1000, so six 0's.
		

Links

• Jaroslav Krizek, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A093653 (number of 1's in binary expansion of all divisors of n).

Cf. A182627 (number of digits in binary expansion of all divisors of n).

Cf. A182621 (concatenation of the divisors of n written in base 2).

Cf. A000217, A000668.

Programs

Mathematica

Table[Count[Flatten[IntegerDigits[Divisors[n], 2]], 0], {n, 81}] (* T. D. Noe, Sep 04 2013 *)

PARI

a(n) = sumdiv(n, d, 1+logint(d, 2) - hammingweight(d)); \\ Michel Marcus, Apr 24 2022

Python

from sympy import divisors
def a(n): return sum(bin(d)[2:].count("0") for d in divisors(n))
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Apr 20 2022

Formula

a(n) = A182627(n) - A093653(n).

a(2^n) = n*(n+1)/2 = A000217(n). - Bernard Schott, Apr 22 2022