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 A065108 Positive numbers expressible as a product of Fibonacci numbers. 13

%I

%S 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,

%T 40,42,45,48,50,52,54,55,60,63,64,65,68,72,75,78,80,81,84,89,90,96,

%U 100,102,104,105,108,110,117,120,125,126,128,130,135,136,144,150,156,160,162

%N Positive numbers expressible as a product of Fibonacci numbers.

%C There are infinitely many triples of consecutive terms of this sequence that are consecutive integers, see A065885. - _John W. Layman_, Nov 27 2001

%C 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

%H T. D. Noe, <a href="/A065108/b065108.txt">Table of n, a(n) for n = 1..10000</a>

%H Clemens Heuberger and Stephan Wagner, <a href="https://arxiv.org/abs/1606.02639">On the monoid generated by a Lucas sequence</a>, arXiv:1606.02639 [math.NT], 2016.

%H David A. Corneth, <a href="/A065108/a065108.gp.txt">Table of 10000*i, a(10000*i), log(a(10000*i))/log(10000*i) for i = 1..470</a>

%F 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

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

%p 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

%t 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 *)

%o (PARI) list(lim)=if(lim<7, return([1..lim\1])); my(v=List(), 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

%Y Cf. A000045, A065885. Complement of A065105.

%Y Cf. A049997 and A094563: F(i)*F(j) and (F(i)*F(j)*F(k) respectively.

%Y Subsequence of A178772.

%K nonn

%O 1,2

%A _Joseph L. Pe_, Nov 21 2001

%E More terms from _John W. Layman_, Nov 27 2001

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

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Last modified October 18 23:39 EDT 2019. Contains 328211 sequences. (Running on oeis4.)