A049998 -id:A049998 - cp's OEIS frontend

A049999 a(n) = smallest index k such that Fibonacci(k) = d(n), where d = A049998 (sequence of first differences of ordered products of Fibonacci numbers, i.e., of A049997, with no duplicates).

1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 3, 1, 5, 4, 1, 1, 6, 5, 1, 3, 7, 6, 1, 1, 4, 8, 7, 3, 1, 5, 9, 8, 4, 1, 1, 6, 10, 9, 5, 1, 3, 7, 11, 10, 6, 1, 1, 4, 8, 12, 11, 7, 3, 1, 5, 9, 13, 12, 8, 4, 1, 1, 6, 10, 14, 13, 9, 5, 1, 3, 7, 11, 15, 14, 10, 6, 1, 1, 4, 8, 12, 16, 15, 11

Offset: 1

Clark Kimberling

nonn

"David W. Wilson conjectured (Dec 14 2005) that" sequence A049998 "consists only of Fibonacci numbers. Proofs were found by Franklin T. Adams-Watters and Don Reble, Dec 14 2005." - Petros Hadjicostas, Nov 08 2019 [This comment was copied from A049998, which includes Don Reble's proof of the conjecture.]

			From _Petros Hadjicostas_, Nov 08 2019: (Start)
A049998(1) = 1 = Fibonacci(1) = Fibonacci(2), so a(1) = min(1,2) = 1.
A049998(7) = 2 = Fibonacci(3), so a(7) = 3.
A049998(10) = 3 = Fibonacci(4), so a(10) = 4.
A049998(13) = 5 = Fibonacci(5), so a(13) = 5.
A049998(17) = 8 = Fibonacci(6), so a(17) = 6. (End)

Clark Kimberling, Orderings of products of Fibonacci numbers, Fibonacci Quarterly 42:1 (2004), pp. 28-35. (Includes a proof of the conjecture proved in the Comments for sequence A049998.)

Cf. A000045, A049997, A049998, A094563, A226857, A271354.

A000045(a(n)) = A049998(n) = A049997(n) - A049997(n-1) for n >= 1. - Petros Hadjicostas, Nov 08 2019

Name edited by and more terms from Petros Hadjicostas, Nov 08 2019

A049997 Numbers of the form Fibonacci(i)*Fibonacci(j).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 15, 16, 21, 24, 25, 26, 34, 39, 40, 42, 55, 63, 64, 65, 68, 89, 102, 104, 105, 110, 144, 165, 168, 169, 170, 178, 233, 267, 272, 273, 275, 288, 377, 432, 440, 441, 442, 445, 466, 610, 699, 712, 714, 715, 720, 754

Offset: 0

Clark Kimberling

It follows from Atanassov et al. that a(n) << sqrt(phi)^n, which matches the trivial a(n) >> sqrt(phi)^n up to a constant factor. - Charles R Greathouse IV, Feb 06 2013

Conjecture: Fibonacci(m)*Fibonacci(n) with 2 < m < n is a perfect power only for (m,n) = (3,6). This has been verified for 2 < m < n <= 900. - Zhi-Wei Sun, Jan 02 2025

			25 is in the sequence since it is the product of two, not necessarily distinct, Fibonacci numbers, 5 and 5.
26 is in the sequence since it is the product of two Fibonacci numbers, 2 and 13.
27 is not in the sequence because there is no way whatsoever to represent it as the product of exactly two Fibonacci numbers.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
K. T. Atanassov, Ron Knott, Kiyota Ozeki, A. G. Shannon, and László Szalay, Inequalities among related pairs of Fibonacci numbers, Fibonacci Quarterly 41:1 (2003), pp. 20-22.
Clark Kimberling, Orderings of products of Fibonacci numbers, Fibonacci Quarterly 42:1 (2004), pp. 28-35.
Zhi-Wei Sun, Perfect powers as products of two Fibonacci or Lucas numbersQuestion 485037 at MathOverflow, December 30, 2024.

Subsequence of A065108; apart from the first term, subsequence of A094563. Complement is A228523.
See A049998 for further information about this sequence. Cf. A080097.
Intersection with A059389 (sums of two Fibonacci numbers) is A226857.
Cf. also A090206, A005478.

Take[ Union@Flatten@Table[ Fibonacci[i]Fibonacci[j], {i, 0, 16}, {j, 0, i}], 61] (* Robert G. Wilson v, Dec 14 2005 *)

list(lim)=my(phi=(1+sqrt(5))/2, v=vector(log(lim*sqrt(5))\log(phi), i, fibonacci(i+1)), u=List([0]),t); for(i=1,#v,for(j=i,#v,t=v[i]*v[j];if(t>lim,break,listput(u,t)))); vecsort(Vec(u),,8) \\ Charles R Greathouse IV, Feb 05 2013

A271356 Difference sequence of the sequence A271354 of the increasing sequence of products of two distinct Fibonacci numbers greater than 1.

Original entry on oeis.org

4, 5, 1, 8, 2, 13, 1, 2, 21, 2, 3, 34, 2, 1, 5, 55, 3, 2, 8, 89, 5, 1, 2, 13, 144, 8, 2, 3, 21, 233, 13, 2, 1, 5, 34, 377, 21, 3, 2, 8, 55, 610, 34, 5, 1, 2, 13, 89, 987, 55, 8, 2, 3, 21, 144, 1597, 89, 13, 2, 1, 5, 34, 233, 2584, 144, 21, 3, 2, 8, 55, 377

Offset: 1

Clark Kimberling, May 02 2016

Conjecture: every term except the first is a Fibonacci number.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A271354, A000045, A049998.

z = 100; f[n_] := Fibonacci[n];
t = Take[Sort[Flatten[Table[f[m] f[n], {n, 3, z}, {m, 3, n - 1}]]], 1000];
Differences[t]

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A049999 a(n) = smallest index k such that Fibonacci(k) = d(n), where d = A049998 (sequence of first differences of ordered products of Fibonacci numbers, i.e., of A049997, with no duplicates).

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A049997 Numbers of the form Fibonacci(i)*Fibonacci(j).

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A271356 Difference sequence of the sequence A271354 of the increasing sequence of products of two distinct Fibonacci numbers greater than 1.

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