A280681 Numbers k such that Fibonacci(k) is a totient.

1, 2, 3, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 72, 84, 90, 96, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 180, 192, 198, 204, 210, 216, 222, 228, 234, 240, 252, 264, 270, 276, 288, 294, 300, 306, 312, 324, 330, 336, 342, 348, 354, 360, 372, 378, 384, 390, 396, 402, 408, 414, 420, 432, 444, 450, 456, 462, 468, 480, 492, 504, 510, 516, 522, 528, 540, 546, 552, 558, 564, 570, 576, 588, 594, 600, 612, 624, 630, 636

Offset

1,2

Comments

Respectively, corresponding Fibonacci numbers are 1, 1, 2, 8, 144, 2584, 46368, 832040, 14930352, 267914296, 4807526976, 86267571272, 1548008755920, 498454011879264, 160500643816367088, 2880067194370816120, ...

Note that sequence does not contain all the positive multiples of 6, e.g., 66 and 102. See A335976 for a related sequence.

Conjecture: Sequence is infinite. - Altug Alkan, Jul 05 2020

All terms > 2 are multiples of 3, because Fibonacci(k) is odd unless k is a multiple of 3. Are all terms > 3 multiples of 6? If a term k is not a multiple of 6, then since Fibonacci(k) is not divisible by 4, Fibonacci(k)+1 must be in A114871. - Robert Israel, Aug 02 2020

Unless there is an odd term > 3, this sequence as a set is {1, 2, 3} U 6*(Z^+ \ A335976). - Max Alekseyev, Dec 08 2024

Links

Max Alekseyev, Table of n, a(n) for n = 1..665

Example

12 is in the sequence because Fibonacci(12) = 144 is in A000010.

Maple

select(k -> numtheory:-invphi(combinat:-fibonacci(k))<>[], [1, 2, seq(i, i=3..100, 3)]); # Robert Israel, Aug 02 2020

Prog

(PARI) isok(k) = istotient(fibonacci(k)); \\ Altug Alkan, Jul 05 2020

Crossrefs

Cf. A000010, A000045, A114871, A280592, A335976.

Sequence in context: A323392 A174801 A324177 * A328899 A093687 A000423

Adjacent sequences: A280678 A280679 A280680 * A280682 A280683 A280684

Keyword

nonn

Author

Altug Alkan, Jan 07 2017

Extensions

a(28)-a(49) from Jinyuan Wang, Jul 08 2020

Terms a(50) onward from Max Alekseyev, Dec 08 2024

Status

approved