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

1,2

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)

Links

Table of n, a(n) for n=1..6.

Formula

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A071295.

Sequence in context: A375919 A156482 A335832 * A319166 A335400 A304193

Adjacent sequences: A173343 A173344 A173345 * A173347 A173348 A173349

Keyword

nonn,base,full,fini

Author

Vladimir Joseph Stephan Orlovsky, Feb 16 2010

Extensions

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

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

Status

approved