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

%I A049997 #40 Jan 05 2025 19:51:35
%S A049997 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,
%T A049997 89,102,104,105,110,144,165,168,169,170,178,233,267,272,273,275,288,
%U A049997 377,432,440,441,442,445,466,610,699,712,714,715,720,754
%N A049997 Numbers of the form Fibonacci(i)*Fibonacci(j).
%C A049997 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
%C A049997 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
%H A049997 Charles R Greathouse IV, <a href="/A049997/b049997.txt">Table of n, a(n) for n = 0..10000</a>
%H A049997 K. T. Atanassov, Ron Knott, Kiyota Ozeki, A. G. Shannon, and László Szalay, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/41-1/atanassov.pdf">Inequalities among related pairs of Fibonacci numbers</a>, Fibonacci Quarterly 41:1 (2003), pp. 20-22.
%H A049997 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/42-1/quartkimberling01_2004.pdf">Orderings of products of Fibonacci numbers</a>, Fibonacci Quarterly 42:1 (2004), pp. 28-35.
%H A049997 Zhi-Wei Sun, <a href="https://mathoverflow.net/questions/485037">Perfect powers as products of two Fibonacci or Lucas numbers</a>Question 485037 at MathOverflow, December 30, 2024.
%e A049997 25 is in the sequence since it is the product of two, not necessarily distinct, Fibonacci numbers, 5 and 5.
%e A049997 26 is in the sequence since it is the product of two Fibonacci numbers, 2 and 13.
%e A049997 27 is not in the sequence because there is no way whatsoever to represent it as the product of exactly two Fibonacci numbers.
%t A049997 Take[ Union@Flatten@Table[ Fibonacci[i]Fibonacci[j], {i, 0, 16}, {j, 0, i}], 61] (* _Robert G. Wilson v_, Dec 14 2005 *)
%o A049997 (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
%Y A049997 Subsequence of A065108; apart from the first term, subsequence of A094563. Complement is A228523.
%Y A049997 See A049998 for further information about this sequence. Cf. A080097.
%Y A049997 Intersection with A059389 (sums of two Fibonacci numbers) is A226857.
%Y A049997 Cf. also A090206, A005478.
%K A049997 nonn,easy
%O A049997 0,3
%A A049997 _Clark Kimberling_