|
|
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
|
|
|
FORMULA
|
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 02 2005
|
|
STATUS
|
approved
|
|
|
|