%I #7 Jun 18 2016 00:30:49
%S 0,1,2,3,5,6,8,10,13,15,16,21,24,26,30,34,39,40,42,48,55,63,65,68,78,
%T 80,89,102,104,105,110,120,126,130,144,165,168,170,178,195,204,208,
%U 210,233,240,267,272,273,275,288,312,315,330,336,340,377,390,432,440,442,445
%N Numbers that are the product of distinct Fibonacci numbers.
%C Starts the same as A049862, the product of two distinct Fibonacci numbers. This sequence has an infinite number of consecutive terms that are consecutive numbers (such as 15 and 16) because fib(k)*fib(k+3) and fib(k+1)*fib(k+2) differ by one for all k >= 0.
%C It follows from Carmichael's theorem that if u and v are finite sets of Fibonacci numbers such that (product of all the numbers in u) = (product of all the numbers in v), then u = v. The same holds for many other 2nd order linear recurrence sequences with constant coefficients. In the following guide to related "distinct product sequences", W = Wythoff array, A035513:
%C base sequence distinct-product sequence
%C A000045 (Fibonacci) A160009
%C A000032 (Lucas, without 2) A274280
%C A000032 (Lucas, with 2) A274281
%C A000285 (1,4,5,...) A274282
%C A022095 (1,5,6,...) A274283
%C A006355 (2,4,6,...) A274284
%C A013655 (2,5,7,...) A274285
%C A022086 (3,6,9,...) A274191
%C row 2 of W: (4,7,11,...) A274286
%C row 3 of W: (6,10,16,...) A274287
%C row 4 of W: (9,15,24,...) A274288
%C - _Clark Kimberling_, Jun 17 2016
%H T. D. Noe, <a href="/A160009/b160009.txt">Table of n, a(n) for n=1..1000</a>
%t s={1}; nn=30; f=Fibonacci[2+Range[nn]]; Do[s=Union[s,Select[s*f[[i]],#<=f[[nn]]&]], {i,nn}]; s=Prepend[s,0]
%Y A059844, A065108
%K nonn
%O 1,3
%A _T. D. Noe_, Apr 29 2009