A064414 Numbers that divide at least one term in every Fibonacci sequence.

1, 2, 3, 4, 6, 7, 9, 14, 23, 27, 43, 49, 67, 81, 83, 86, 98, 103, 127, 134, 163, 167, 206, 223, 227, 243, 254, 283, 326, 343, 367, 383, 443, 446, 463, 467, 487, 503, 523, 529, 547, 566, 587, 607, 643, 647, 683, 686, 727, 729, 734, 787, 823, 827, 863, 883, 887

Offset

1,2

Comments

Previous name was: Fix a > 0, b > 0, k > 0 and define G_1 = a, G_2 = b, G_k = G_(k-1) + G_(k-2); sequence gives m such that for any (a,b), some G_k is divisible by m.

From Logan J. Kleinwaks, Oct 29 2017: (Start)

The squares of this sequence are the squares in A232656.

Conjecture: these are the numbers j such that j^2 = Sum_{d|j} phi(d)*A001177(d), where phi = Euler's totient function (A000010). See A232656. (End)

References

Teruo Nishiyama, Fibonacci numbers, Suuri-Kagaku, No. 285, March 1987, 67-69, (in Japanese).

Links

Daniel Mondot, Table of n, a(n) for n = 1..10000 (first 360 terms from Amiram Eldar)

Brandon Avila and Tanya Khovanova, Free Fibonacci Sequences, Journal of Integer Sequences, Vol. 17 (2014), Article 14.8.5; arXiv preprint, arXiv:1403.4614 [math.NT], 2014.

Example

If a = 1, b = 4, then G_k is (1, 4, 5, 9, 14, 23, ...) and no G_k is a multiple of 11. Therefore 11 is not in the sequence.

Mathematica

g[a_, b_, k_] := Fibonacci[k-2]*a + Fibonacci[k-1]*b; ok[n_] := Catch[ Do[ test = Catch[ Do[ If[ Divisible[g[a, b, k], n], Throw[True]], {k, 1, 2*n}]]; If[test == Null, Throw[False]], {a, 1, Floor[Sqrt[n]]}, {b, 1, Floor[Sqrt[n]]}]] ; Reap[ Do[ If[ok[n] == Null, Print[n]; Sow[n]], {n, 1, 1000}]][[2, 1]] (* Jean-François Alcover, Jul 19 2012 *)

Crossrefs

Prime terms are in A000057.

Composite terms are in A389401.

Cf. A232357, A232656.

Sequence in context: A055494 A239115 A165773 * A224482 A002475 A208281

Adjacent sequences: A064411 A064412 A064413 * A064415 A064416 A064417

Keyword

easy,nonn,nice

Author

Naohiro Nomoto, Oct 15 2001

Extensions

More terms from David Wasserman, Jul 18 2002

Name edited by David A. Corneth, Oct 30 2017

Name edited by Daniel Mondot, Sep 28 2025

Status

approved