A214560 Number of 0's in binary expansion of n^2.

1, 0, 2, 2, 4, 2, 4, 3, 6, 4, 4, 2, 6, 4, 5, 4, 8, 6, 6, 4, 6, 3, 4, 7, 8, 5, 6, 4, 7, 5, 6, 5, 10, 8, 8, 6, 8, 5, 6, 4, 8, 6, 5, 4, 6, 3, 9, 8, 10, 7, 7, 7, 8, 4, 6, 5, 9, 6, 7, 5, 8, 6, 7, 6, 12, 10, 10, 8, 10, 7, 8, 6, 10, 7, 7, 4, 8, 6, 6, 8, 10, 7, 8, 5, 7

Offset: 0

Author

Alex Ratushnyak, Jul 21 2012

Keywords

• base
• nonn

Comments

Conjecture: for every x>=0 there is an i such that a(n)>x for n>i.

Comment from N. J. A. Sloane, Nov 21 2013: See also the conjecture in A231898.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000120, A000290, A023416, A078565, A159918, A231898.

Programs

Haskell

a214560 = a023416 . a000290  -- Reinhard Zumkeller, Nov 20 2013

Maple

A214560 := proc(n)
    A023416(n^2) ;
end proc: # R. J. Mathar, Jul 21 2012
# second Maple program:
a:= n-> `if`(n=0, 1, add(1-i, i=Bits[Split](n^2))):
seq(a(n), n=0..84);  # Alois P. Heinz, Nov 25 2024

Mathematica

Join[{1},Table[DigitCount[n^2,2,0],{n,100}]] (* Harvey P. Dale, Nov 24 2024 *)

PARI

vector(66,n,b=binary((n-1)^2);sum(j=1,#b,1-b[j])) /* Joerg Arndt, Jul 21 2012 */

Python

for n in range(300):
    b = n*n
    c = 0
    while b>0:
        c += 1-(b&1)
        b//=2
    print(c+(n==0), end=', ')

Python

def A214560(n):
    return bin(n*n)[2:].count('0') # Chai Wah Wu, Sep 03 2014

Formula

a(n) = A023416(A000290(n)).