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A226590
Total number of 0's in binary expansion of all divisors of n.
3
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
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).
Sequence in context: A318526 A054869 A201671 * A261349 A227962 A331105
KEYWORD
base,nonn
AUTHOR
Jaroslav Krizek, Aug 31 2013
STATUS
approved