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 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 n such there exists (a, b) where G_k is divisible by n. 6
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Conjecture: these are the n such that n^2 = Sum_{d|n} phi(d)*A001177(d), where phi = Euler's totient function (A000010). See A232656. - Logan J. Kleinwaks, Oct 29 2017 REFERENCES Teruo Nishiyama, Fibonacci numbers, Suuri-Kagaku, No. 285, March 1987, 67-69, (in Japanese). LINKS B. Avila and T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and J. Int. Seq. 17 (2014) # 14.8.5. 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 members are in A000057. The squares of this sequence are the squares in A232656. - Logan J. Kleinwaks, Oct 29 2017 Cf. A232357. 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

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Last modified December 9 00:32 EST 2019. Contains 329871 sequences. (Running on oeis4.)