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

1,3

Comments

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

Links

Table of n, a(n) for n=1..60.

Example

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.

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 *)

Crossrefs

Cf. A000045, A381704.

Sequence in context: A053433 A091401 A278581 * A191889 A091402 A365513

Adjacent sequences: A382050 A382051 A382052 * A382054 A382055 A382056

Keyword

nonn,base

Author

Robert Israel, Mar 13 2025

Status

approved