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Number of 0's in binary expansion of n^2.
5

%I #24 Sep 03 2014 23:46:03

%S 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,

%T 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,

%U 10,8,10,7,8,6,10,7,7,4,8,6,6,8,10,7,8,5,7

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

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

%C Comment from _N. J. A. Sloane_, Nov 21 2013: See also the conjecture in A231898.

%H Reinhard Zumkeller, <a href="/A214560/b214560.txt">Table of n, a(n) for n = 0..10000</a>

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

%p A214560 := proc(n)

%p A023416(n^2) ;

%p end proc: # _R. J. Mathar_, Jul 21 2012

%o (Python)

%o for n in range(300):

%o b = n*n

%o c = 0

%o while b>0:

%o c += 1-(b&1)

%o b/=2

%o print c+(n==0),

%o (PARI) vector(66,n,b=binary((n-1)^2);sum(j=1,#b,1-b[j])) /* _Joerg Arndt_, Jul 21 2012 */

%o (Haskell)

%o a214560 = a023416 . a000290 -- _Reinhard Zumkeller_, Nov 20 2013

%o (Python)

%o def A214560(n):

%o ....return bin(n*n)[2:].count('0') # _Chai Wah Wu_, Sep 03 2014

%Y Cf. A000120, A023416, A078565, A159918, A231898.

%K base,nonn

%O 0,3

%A _Alex Ratushnyak_, Jul 21 2012