

A064414


Fix a > 0, b > 0, k > 0 and define G_1 = a, G_2 = b, G_k = G_(k1) + G_(k2); 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
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OFFSET

1,2


COMMENTS

Conjecture: these are the n such that n^2 = Sum_{dn} phi(d)*A001177(d), where phi = Euler's totient function (A000010). See A232656.  Logan J. Kleinwaks, Oct 29 2017


REFERENCES

Teruo Nishiyama, Fibonacci numbers, SuuriKagaku, No. 285, March 1987, 6769, (in Japanese).


LINKS

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.


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[k2]*a + Fibonacci[k1]*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]] (* JeanFrançois 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



