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

1,4

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

Links

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

Formula

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

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

Example

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

Mathematica

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

Prog

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

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.

Sequence in context: A318526 A054869 A201671 * A261349 A227962 A331105

Adjacent sequences: A226587 A226588 A226589 * A226591 A226592 A226593

Keyword

base,nonn

Author

Jaroslav Krizek, Aug 31 2013

Status

approved