A232357 -id:A232357 - cp's OEIS frontend

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

Naohiro Nomoto, Oct 15 2001

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)

			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.

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

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.

Prime terms are in A000057.

Cf. A232357, A232656.

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 *)

More terms from David Wasserman, Jul 18 2002

Name edited by David A. Corneth, Oct 30 2017

A232656 The number of pairs of numbers below n that, when generating a Fibonacci-like sequence modulo n, contain zeros.

Original entry on oeis.org

1, 4, 9, 16, 21, 36, 49, 40, 81, 84, 101, 96, 85, 196, 189, 136, 145, 180, 325, 336, 153, 404, 529, 216, 521, 340, 729, 496, 393, 756, 901, 520, 509, 292, 1029, 384, 685, 652, 765, 840, 801, 612, 1849, 1016, 1701, 1060, 737, 504, 2401, 2084, 1305, 1360, 1405, 1476, 521, 1096, 1629, 1572

Offset: 1

Brandon Avila and Tanya Khovanova, Nov 27 2013

nonn

a(n) = n^2 iff n is in A064414, a(n) is not equal to n^2 iff n is in A230457.

a(n) + A232357(n) = n^2.

			The sequence 2,1,3,4,2,1 is the sequence of Lucas numbers modulo 5. Lucas numbers are never divisible by 5. The 4 pairs (2,1), (1,3), (3,4), (4,2) are the only pairs that can generate a sequence modulo 5 that doesn't contain zeros. Thus, a(5) = 21, as 21 other pairs generate sequences that do contain zeros.
Any Fibonacci like sequence contains elements divisible by 2, 3, or 4. Thus, a(2) = 4, a(3) = 9, a(4) = 16.

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.

Cf. A064414, A230457, A232357.

fibLike[list_] := Append[list, list[[-1]] + list[[-2]]]; Table[k^2 -Count[Flatten[Table[Count[Nest[fibLike, {n, m}, k^2]/k, _Integer], {n, k - 1}, {m, k - 1}]], 0], {k, 70}]

Conjecture: a(n) = Sum_{d|n} phi(d)*A001177(d), where phi = Euler's totient function (A000010). - Logan J. Kleinwaks, Oct 28 2017

a(n) = Sum_{d|n} phi(d)*A001177(d) = Sum_{k=1..n} A001177(n/gcd(n,k)) = Sum_{k=1..n} A001177(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 09 2021

cp's OEIS Frontend

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.

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