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A280681
Numbers k such that Fibonacci(k) is a totient.
2
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
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
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
KEYWORD
nonn,more
AUTHOR
Altug Alkan, Jan 07 2017
EXTENSIONS
a(28)-a(49) from Jinyuan Wang, Jul 08 2020
STATUS
approved