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A065108 Positive numbers expressible as a product of Fibonacci numbers. 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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
LINKS
Clemens Heuberger and Stephan Wagner, On the monoid generated by a Lucas sequence, arXiv:1606.02639 [math.NT], 2016.
FORMULA
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
EXAMPLE
52 = 2 * 2 * 13 is the product of Fibonacci numbers 2, 2 and 13.
MAPLE
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
MATHEMATICA
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 *)
PROG
(PARI) 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
CROSSREFS
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.
Sequence in context: A168134 A245030 A245027 * A094563 A228897 A068095
KEYWORD
nonn
AUTHOR
Joseph L. Pe, Nov 21 2001
EXTENSIONS
More terms from John W. Layman, Nov 27 2001
More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 02 2005
STATUS
approved

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Last modified March 28 17:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)