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%I #16 Sep 21 2021 21:31:08
%S 0,4,16,144,324,625
%N 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.
%C From _Rémy Sigrist_, Apr 30 2017: (Start)
%C In binary:
%C - the product of numbers of 0's and 1's for an N-digit number is at most N^2/4,
%C - the least N-digit number is 2^(N-1),
%C - for N >= 11, (N^2/4)^2 < 2^(N-1).
%C Hence there are no terms >= 2^10.
%C (End)
%F Terms satisfy m = A071295(m)^2. - _Michel Marcus_, Nov 19 2015
%e 625 -> 1001110001; five '0' and five '1'; 5*5=25; sqrt(625)=25.
%e 324 -> 101000100; 3 '0' and 6 '1'; 3*6=18; sqrt(324)=18.
%t Select[Range[8! ],DigitCount[ #,2,0]*DigitCount[ #,2,1]==Sqrt[ # ]&]
%o (PARI) isok(n) = {n1 = hammingweight(n); n0 = #binary(n) - n1; (n0*n1)^2 == n;} \\ _Michel Marcus_, Nov 19 2015
%Y Cf. A071295.
%K nonn,base,full,fini
%O 1,2
%A _Vladimir Joseph Stephan Orlovsky_, Feb 16 2010
%E Minor edits by _N. J. A. Sloane_, Feb 21 2010
%E a(1) = 0 inserted by _Michel Marcus_, Nov 19 2015