A173346 Numbers such that the product of numbers of 0's and 1's in the binary representation is equal to the square root of the number.

0, 4, 16, 144, 324, 625

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Feb 16 2010

Keywords

• nonn
• base
• full
• fini

Comments

From Rémy Sigrist, Apr 30 2017: (Start)

In binary:

- the product of numbers of 0's and 1's for an N-digit number is at most N^2/4,

- the least N-digit number is 2^(N-1),

- for N >= 11, (N^2/4)^2 < 2^(N-1).

Hence there are no terms >= 2^10.

(End)

Examples

			625 -> 1001110001; five '0' and five '1'; 5*5=25; sqrt(625)=25.
324 -> 101000100; 3 '0' and 6 '1'; 3*6=18; sqrt(324)=18.
		

Crossrefs

Cf. A071295.

Programs

Mathematica

Select[Range[8! ],DigitCount[ #,2,0]*DigitCount[ #,2,1]==Sqrt[ # ]&]

PARI

isok(n) =  {n1 = hammingweight(n); n0 = #binary(n) - n1; (n0*n1)^2 == n;} \\ Michel Marcus, Nov 19 2015

Formula

Terms satisfy m = A071295(m)^2. - Michel Marcus, Nov 19 2015

Extensions

Minor edits by N. J. A. Sloane, Feb 21 2010

a(1) = 0 inserted by Michel Marcus, Nov 19 2015