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A049862 Products of two Fibonacci numbers with distinct indices.

%I A049862 #36 Jan 05 2025 19:51:35
%S A049862 0,1,2,3,5,6,8,10,13,15,16,21,24,26,34,39,40,42,55,63,65,68,89,102,
%T A049862 104,105,110,144,165,168,170,178,233,267,272,273,275,288,377,432,440,
%U A049862 442,445,466,610,699,712,714,715,720,754,987,1131
%N A049862 Products of two Fibonacci numbers with distinct indices.
%C A049862 There are no duplicates except for the trivial cases 1*F(j)=1*F(j) and F(i)*F(j)=F(j)*F(i). - _Robert Israel_, May 11 2016
%C A049862 The number 1 is included because 1 = F(1)*F(2). - _Clark Kimberling_, Jun 19 2016
%H A049862 T. D. Noe, <a href="/A049862/b049862.txt">Table of n, a(n) for n=1..1000</a>
%H A049862 Mohammad K. Azarian, The Value of a Series of Reciprocal Fibonacci Numbers, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/ElemProbAugust2013.pdf">Problem B-1133</a>, Fibonacci Quarterly, Vol. 51, No. 3, August 2013, p. 275. <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/ElemProbSolnAug14.pdf">Solution</a> published in Vol. 52, No. 3, August 2014, pp. 277-278.
%H A049862 MathOverflow, <a href="http://mathoverflow.net/questions/238505/distinctness-of-products-of-fibonacci-numbers/238508#238508">Distinctness of products of Fibonacci numbers</a>
%p A049862 fib:= combinat:-fibonacci:
%p A049862 sort(convert(select(`<`,{0,seq(seq(fib(i)*fib(j),i=j+1..100),j=1..100)},fib(101)),list)); # _Robert Israel_, May 11 2016
%t A049862 Take[Union[Flatten[Table[Fibonacci[i]*Fibonacci[j], {i, 0, 100}, {j, i + 1, 100}]]], 100] (* _Clark Kimberling_, May 11 2016 *)
%o A049862 (PARI) isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
%o A049862 isok(n) = {if ((n==0) || (n==1), return (1)); fordiv(n, d, if (d^2 < n, if (isfib(d) && isfib(n/d), return (1)););); return(0);} \\ _Michel Marcus_, May 27 2019
%o A049862 (PARI) lista(nn) = {my(out = List([0])); for (i=0, nn, for (j=i+1, nn, listput(out, fibonacci(i)*fibonacci(j)););); Vec(vecsort(select(x->(x < fibonacci(nn+1)), out), , 8));} \\ _Michel Marcus_, May 27 2019
%Y A049862 Cf. A000045, A160009, A272949.
%K A049862 nonn
%O A049862 1,3
%A A049862 _Clark Kimberling_
%E A049862 Name changed to conform with A272949 et al. by _Clark Kimberling_, Jun 18 2016