A280592 -id:A280592 - cp's OEIS frontend

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

Altug Alkan, Jan 07 2017

nonn

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

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

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

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

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

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

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

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

A335976 Numbers k such that Fibonacci(6*k) is not a totient.

Original entry on oeis.org

0, 11, 13, 17, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 113, 121, 131, 137, 139, 141, 149, 151, 157, 167, 173, 191, 193, 199, 223, 229, 233, 239, 241, 243, 257, 263, 271, 281, 283, 293, 311, 313, 317, 321, 331, 339, 347, 349, 353, 373, 389, 397, 401, 419, 421, 431, 433, 443, 449, 457, 461, 479, 487, 509, 521, 541, 557, 573, 577, 587, 599, 613, 617, 619, 631, 641, 643, 653, 661, 673, 733, 739

Offset: 1

Altug Alkan, Jul 03 2020

nonn

Conjecture: Sequence contains infinitely many primes.

			11 is a term since Fibonacci(66) = 27777890035288 is not a totient number.

Cf. A000010, A007617, A134492, A280592, A280681.

PARI
```
isok(n) = !istotient(fibonacci(6*n))
```

a(12)-a(20) from Max Alekseyev, Aug 02 2020

Terms a(21) onward from Max Alekseyev, May 19 2024

A373194 Numbers k such that phi(k) is a Lucas number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 19, 27, 38, 54, 2049, 2732, 4098, 5779, 11558, 36717, 48956, 73434, 21994424093409, 29325898791212, 43988848186818, 439894502304193355596420713117, 586526003072257807461894284156, 879789004608386711192841426234, 56570478046795035524653081529155199270281, 56570478046795035532692004624509431078281

Offset: 1

Darío Clavijo, May 27 2024

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..64
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A000032, A102460, A280592.

lucasQ[n_] := Or @@ (IntegerQ[Sqrt[#]] & /@ (5*n^2 + 20*{-1, 1})); Select[Range[10^4], lucasQ[EulerPhi[#]] &] (* Amiram Eldar, May 27 2024 *)

isok(k) = islucas(eulerphi(k)); \\ using islucas from A102460 \\ Michel Marcus, May 27 2024

\\ read Max Alekseyev's invphi.gp
a373194(uptoNLucas) = my(A=List()); for(n=0, uptoNLucas, my(L = invphi(fibonacci(n+1) + fibonacci(n-1))); if(#L, for(k=1, #L, listput(A,L[k])))); Set(A);
a373194(150) \\ Hugo Pfoertner, Jun 10 2024

from sympy.ntheory.primetest import is_square
from sympy import totient
islucas = lambda n: is_square(5*n*n - 20) or is_square(5*n*n + 20)
print([n for n in range(1,10**4) if islucas(totient(n))])

a(18)-a(21) from Amiram Eldar, May 27 2024

a(22) onwards from Hugo Pfoertner, May 27 2024

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

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A335976 Numbers k such that Fibonacci(6*k) is not a totient.

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A373194 Numbers k such that phi(k) is a Lucas number.

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