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Biased numbers: n such that all terms of the sequence f(n), f(f(n)), f(f(f(n))), ..., 1, where f(k) = floor(k/2), are odd.
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%I #16 Dec 27 2023 20:41:07

%S 2,3,6,7,14,15,30,31,62,63,126,127,254,255,510,511,1022,1023,2046,

%T 2047,4094,4095,8190,8191,16382,16383,32766,32767,65534,65535,131070,

%U 131071,262142,262143,524286,524287,1048574,1048575,2097150,2097151,4194302,4194303

%N Biased numbers: n such that all terms of the sequence f(n), f(f(n)), f(f(f(n))), ..., 1, where f(k) = floor(k/2), are odd.

%C This sequence consists of all numbers of the form 2^k - 2, 2^k - 1, where k >= 2.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-2).

%F From _Alois P. Heinz_, Dec 27 2023: (Start)

%F G.f.: -x*(2*x^3-3*x-2)/((x-1)*(x+1)*(2*x^2-1)).

%F a(n) = 2^floor((n+3)/2)-1-(n mod 2). (End)

%e The sequence corresponding to 14 is 7, 3, 1, all of whose terms are odd. So 14 is a term of the sequence.

%t atsoQ[n_]:=AllTrue[Rest[NestWhileList[Floor[#/2]&,n,#>1&]],OddQ]; Select[Range[2,42*10^5],atsoQ] (* _Harvey P. Dale_, Dec 27 2023 *)

%Y Cf. A075427.

%K easy,nonn

%O 1,1

%A _Joseph L. Pe_, Jan 21 2002

%E More terms from Larry Reeves (larryr(AT)acm.org), Jun 11 2002