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A065108 Numbers expressible as a product of Fibonacci numbers. 9
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

T. D. Noe, Table of n, a(n) for n = 1..10000

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 *)

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

Adjacent sequences:  A065105 A065106 A065107 * A065109 A065110 A065111

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 December 3 10:38 EST 2016. Contains 278699 sequences.