A065108 Positive numbers expressible as a product of Fibonacci numbers.

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

Offset: 1

Author

Joseph L. Pe, Nov 21 2001

Keywords

• nonn

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

Examples

			52 = 2 * 2 * 13 is the product of Fibonacci numbers 2, 2 and 13.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Wawrzyniec Bieniawski, Piotr Masierak, Andrzej Tomski, and Szymon Łukaszyk, Assembly Theory - Formalizing Assembly Spaces and Discovering Patterns and Bounds, Preprints.org (2025).
• Clemens Heuberger and Stephan Wagner, On the monoid generated by a Lucas sequence, arXiv:1606.02639 [math.NT], 2016.
• David A. Corneth, Table of 10000*i, a(10000*i), log(a(10000*i))/log(10000*i) for i = 1..470

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.

Programs

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 *)

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

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

Extensions

More terms from John W. Layman, Nov 27 2001

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 02 2005