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A160009 Numbers that are the product of distinct Fibonacci numbers.

%I A160009 #7 Jun 18 2016 00:30:49
%S A160009 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 A160009 80,89,102,104,105,110,120,126,130,144,165,168,170,178,195,204,208,
%U A160009 210,233,240,267,272,273,275,288,312,315,330,336,340,377,390,432,440,442,445
%N A160009 Numbers that are the product of distinct Fibonacci numbers.
%C A160009 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 A160009 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 A160009 base sequence              distinct-product sequence
%C A160009 A000045 (Fibonacci)             A160009
%C A160009 A000032 (Lucas, without 2)      A274280
%C A160009 A000032 (Lucas, with 2)         A274281
%C A160009 A000285 (1,4,5,...)             A274282
%C A160009 A022095 (1,5,6,...)             A274283
%C A160009 A006355 (2,4,6,...)             A274284
%C A160009 A013655 (2,5,7,...)             A274285
%C A160009 A022086 (3,6,9,...)             A274191
%C A160009 row 2 of W: (4,7,11,...)        A274286
%C A160009 row 3 of W: (6,10,16,...)       A274287
%C A160009 row 4 of W: (9,15,24,...)       A274288
%C A160009 - _Clark Kimberling_, Jun 17 2016
%H A160009 T. D. Noe, <a href="/A160009/b160009.txt">Table of n, a(n) for n=1..1000</a>
%t A160009 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 A160009 A059844, A065108
%K A160009 nonn
%O A160009 1,3
%A A160009 _T. D. Noe_, Apr 29 2009