A094563 -id:A094563 - cp's OEIS frontend

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

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

A065108 Positive numbers expressible as a product of Fibonacci numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 16, 18, 20, 21, 24, 25, 26, 27, 30, 32, 34, 36, 39, 40, 42, 45, 48, 50, 52, 54, 55, 60, 63, 64, 65, 68, 72, 75, 78, 80, 81, 84, 89, 90, 96, 100, 102, 104, 105, 108, 110, 117, 120, 125, 126, 128, 130, 135, 136, 144, 150, 156, 160, 162

Offset: 1

Joseph L. Pe, Nov 21 2001

nonn

There are infinitely many triples of consecutive terms of this sequence that are consecutive integers, see A065885. - John W. Layman, Nov 27 2001

Carmichael's theorem implies that 8 and 144 are the only Fibonacci numbers that are products of other Fibonacci numbers, cf. A235383. - Robert C. Lyons, Jan 13 2013

			52 = 2 * 2 * 13 is the product of Fibonacci numbers 2, 2 and 13.

T. D. Noe, Table of n, a(n) for n = 1..10000
Wawrzyniec Bieniawski, Piotr Masierak, Andrzej Tomski, and Szymon Łukaszyk, Assembly Theory - Formalizing Assembly Spaces and Discovering Patterns and Bounds, Preprints.org (2025).
Clemens Heuberger and Stephan Wagner, On the monoid generated by a Lucas sequence, arXiv:1606.02639 [math.NT], 2016.
David A. Corneth, Table of 10000*i, a(10000*i), log(a(10000*i))/log(10000*i) for i = 1..470

Cf. A000045, A065885. Complement of A065105.
Cf. A049997 and A094563: F(i)*F(j) and F(i)*F(j)*F(k) respectively.
Subsequence of A178772.

with(combinat): A000045:=proc(n) options remember: RETURN(fibonacci(n)): end: mulfib:=proc(m,i) local j,q,f: f:=0: for j from i by -1 to 3 while(f=0) do if(irem(m, A000045(j))=0) then q:=iquo(m, A000045(j)): if(q=1) then RETURN(1) else f:=mulfib(q,j) fi fi od: RETURN(f): end: for i from 3 to 12 do for n from A000045(i) to A000045(i+1)-1 do m:=mulfib(n,i): if m=1 then printf("%d, ",n) fi od od: # C. Ronaldo

nn = 1000; k = 1; fib = {}; While[k++; f = Fibonacci[k]; f <= nn, AppendTo[fib, f]]; s = fib; While[s2 = Select[Union[s, Flatten[Outer[Times, fib, s]]], # <= nn &]; Length[s2] > Length[s], s = s2]; s (* T. D. Noe, Jul 17 2012 *)

list(lim)=if(lim<7, return([1..lim\1])); my(v=List([1]), F=List([2,3]), curfib, t, idx, newidx); while((t=F[#F]+F[#F-1])<=lim, listput(F,t)); F=setminus(Set(F), [8,144]); for(i=1,#F, curfib=F[i]; idx=1; while(v[idx]*curfib<=lim, newidx=#v+1; for(j=idx,#v, t=curfib*v[j]; if(t<=lim, listput(v,t))); idx=newidx)); Set(v) \\ Charles R Greathouse IV, Jun 15 2017

As Charles R Greathouse IV recently remarked, it would be good to have an asymptotic formula for this sequence. - N. J. A. Sloane, Jul 22 2012

More terms from John W. Layman, Nov 27 2001

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 02 2005

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).

Original entry on oeis.org

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

A094564 Triple products of distinct Fibonacci numbers: F(i)F(j)F(k), 2<=i

Original entry on oeis.org

6, 10, 15, 16, 24, 26, 30, 39, 40, 42, 48, 63, 65, 68, 78, 80, 102, 104, 105, 110, 120, 126, 130, 165, 168, 170, 178, 195, 204, 208, 210, 267, 272, 273, 275, 288, 312, 315, 330, 336, 340, 432, 440, 442, 445, 466, 504, 510, 520, 534, 544, 546, 550, 699, 712, 714

Offset: 1

Clark Kimberling, May 12 2004

nonn

			F(2)F(3)F(4)=6 < F(2)F(3)F(5)=10 < F(3)F(4)F(5)=30 < ...

Clark Kimberling, Orderings of products of Fibonacci numbers, Fibonacci Quarterly 42:1 (2004), pp. 28-35.

Cf. A000045, A094563.

With[{nn=30},Take[Union[Times@@@Subsets[Fibonacci[Range[2,nn]],{3}]], 2*nn]] (* Harvey P. Dale, Dec 23 2015 *)

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

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A065108 Positive numbers expressible as a product of Fibonacci numbers.

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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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A094564 Triple products of distinct Fibonacci numbers: F(i)F(j)F(k), 2<=i

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