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

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

0,3

Comments

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

Links

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.

Example

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.

Mathematica

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

Prog

(PARI) 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

Crossrefs

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.

Sequence in context: A375204 A193299 A022773 * A226857 A256007 A324689

Adjacent sequences: A049994 A049995 A049996 * A049998 A049999 A050000

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved