A064414 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 numbers m such that there exists (a, b) where G_k is divisible by m.

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

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

Amiram Eldar, Table of n, a(n) for n = 1..360

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.

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

Status

approved