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

Author

Altug Alkan, Jan 07 2017

Keywords

• nonn

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

Examples

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

Links

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

Crossrefs

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

Programs

Maple

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

PARI

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

Extensions

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

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