A382053 Numbers k such that Fibonacci(k) has a Fibonacci number of 1's in its binary representation.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 19, 20, 22, 30, 33, 46, 47, 56, 85, 105, 109, 150, 173, 254, 266, 279, 413, 416, 444, 624, 651, 690, 713, 746, 1031, 1110, 2841, 2864, 2867, 2892, 2895, 2994, 4516, 4523, 4543, 4559, 7452, 7491, 7532, 11840, 11852, 11863, 19297, 19311, 19442, 19462

Offset: 1

Author

Robert Israel, Mar 13 2025

Keywords

• nonn
• base

Comments

Numbers k such that A000045(k) is in A381704.

Examples

			a(11) = 10 is a term because Fibonacci(10) = 55 = 110111_2 has 5 1's in its binary representation, and 5 = Fibonacci(5) is a Fibonacci number.
		

Crossrefs

Cf. A000045, A381704.

Programs

Maple

isfib:= n -> issqr(5*n^2+4) or issqr(5*n^2-4);
filter:= n -> isfib(convert(convert(combinat:-fibonacci(n),base,2),`+`)):
select(filter, [$0..20000]);

Mathematica

Select[Range[0,20000],ResourceFunction["FibonacciQ"][Total[IntegerDigits[Fibonacci[#],2]]]&] (* or if ResourceFunction Add-on is not available *) Select[Range[0, 20000],AnyTrue[Sqrt[5 #^2 + 4 {-1, 1}] &[DigitSum[Fibonacci[#], 2]],IntegerQ] &] (* James C. McMahon, Mar 14 2025 *)