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A064414
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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.
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6
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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
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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Teruo Nishiyama, Fibonacci numbers, Suuri-Kagaku, No. 285, March 1987, 67-69, (in Japanese).
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LINKS
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Table of n, a(n) for n=1..57.
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.
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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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
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KEYWORD
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easy,nonn,nice
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AUTHOR
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Naohiro Nomoto, Oct 15 2001
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EXTENSIONS
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More terms from David Wasserman, Jul 18 2002
Name edited by David A. Corneth, Oct 30 2017
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STATUS
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approved
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