A178772 -id:A178772 - cp's OEIS frontend

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

Joseph L. Pe, Nov 21 2001

nonn

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

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

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

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.

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

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 *)

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

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

More terms from John W. Layman, Nov 27 2001

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

A178777 Numbers that are not Fibonacci integers.

Original entry on oeis.org

37, 43, 53, 59, 67, 71, 73, 74, 79, 83, 86, 97, 101, 103, 106, 109, 111, 113, 118, 127, 129, 131, 134, 137, 139, 142, 146, 148, 149, 151, 157, 158, 159, 163, 166, 167, 172, 173, 177, 179, 181, 185, 191, 193, 194, 197, 201, 202, 206, 212, 213, 215, 218, 219

Offset: 1

T. D. Noe, Jun 11 2010

nonn

T. D. Noe, Table of n, a(n) for n=1..1000

Complement of A178772.

A201010 Integers that can be written as the product and/or quotient of Lucas numbers.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 23, 24, 27, 28, 29, 31, 32, 33, 36, 38, 41, 42, 44, 46, 47, 48, 49, 54, 56, 57, 58, 62, 63, 64, 66, 69, 72, 76, 77, 81, 82, 84, 87, 88, 92, 93, 94, 96, 98, 99, 107, 108, 112, 114, 116, 121, 123, 124, 126, 128

Offset: 1

Arkadiusz Wesolowski, Jan 08 2013

These numbers do not occur in A178777.
The first 20 terms of this sequence are the same as in A004144 (nonhypotenuse numbers).
Integers of the form A200381(n)/A200381(m) for some m and n.

			19 is in the sequence because Lucas(9)/Lucas(0)^2 = 19.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Lucas Number
Index entries for Lucas sequences

Cf. A000032, A200381, A200995, A201011. Subsequence of A178772. Complement of A201012.

maxTerm = 128; Clear[f]; f[lim_] := f[lim] = (luc = LucasL[Range[0, lim]]; luc = Delete[luc, 2];  last = luc[[-1]]; t = {1}; Do[t2 = luc[[n]]^Range[ Floor[ Log[last] / Log[ luc[[n]] ]]]; s = Select[ Union[ Flatten[ Outer[ Times, t, t2]]], # <= last &]; t = Union[t, s], {n, lim}]; maxIndex = Length[A200381 = t]; Reap[ Do[r = A200381[[n]] / A200381[[m]]; If[IntegerQ[r] && r <= maxTerm, Sow[r]], {n, 1, maxIndex}, {m, 1, maxIndex}]][[2, 1]] // Union); f[5]; f[lim = 10]; While[ Print["lim = ", lim]; f[lim] != f[lim-5], lim = lim+5]; f[lim] (* Jean-François Alcover, Jun 24 2015, after script by T. D. Noe in A200381 *)

A185060 Number of Fibonacci integers in the interval [1, 10^n].

Original entry on oeis.org

10, 88, 534, 2645, 11254, 42735

Offset: 1

Arkadiusz Wesolowski, Dec 25 2012

A Fibonacci integer is an integer in the multiplicative group generated by the Fibonacci numbers. For each fixed epsilon > 0,
exp(C*(log(10^n))^1/2 - (log(10^n))^epsilon) < a(n) < exp(C*(log(10^n))^1/2 + (log(10^n))^(1/6+epsilon)) for x sufficiently large, where C = 2*zeta(2)*sqrt(zeta(3)/(zeta(6)*log((1 + sqrt(5))/2))) = 5.15512.... (Luca, Pomerance, Wagner (2010))
The old entry a(4) = 2681 was the result of an incorrect calculation by Luca, Pomerance and Wagner. - Arkadiusz Wesolowski, Feb 05 2013

Florian Luca, Carl Pomerance, and Stephen Wagner, Fibonacci integers (Banff conference in honor of Cam Stewart, May 31, 2010 to June 4, 2010.)

Cf. A178772, A178777.

e = 4; (*lst1=the terms of A178762 that are smaller than 10^e*); lst1 = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 61, 89, 107, 199, 211, 233, 281, 421, 521, 1103, 1597, 2161, 2207, 2521, 3001, 3571, 5779, 9349, 9901}; lst2 = {}; q = Times @@ Complement[Prime@Range[10^e], lst1]; Do[If[GCD[q, n] == 1, AppendTo[lst2, n]], {n, 10^e}]; Table[Length@Select[lst2, # <= 10^d &], {d, e}] (* Arkadiusz Wesolowski, Feb 05 2013 *)

a(4) corrected by T. D. Noe and Arkadiusz Wesolowski, Feb 05 2013

a(5)-a(6) from Arkadiusz Wesolowski, Feb 06 2013

A275976 Decimal expansion of a constant relating to the density of Fibonacci integers.

Original entry on oeis.org

5, 1, 5, 5, 1, 2, 4, 3, 4, 0, 0, 7, 4, 6, 4, 4, 0, 5, 5, 1, 4, 1, 6, 1, 9, 3, 3, 7, 5, 6, 5, 2, 2, 8, 2, 8, 7, 4, 8, 5, 7, 6, 0, 4, 5, 1, 8, 8, 1, 1, 0, 0, 2, 4, 8, 3, 1, 4, 3, 1, 1, 0, 7, 7, 6, 9, 7, 3, 5, 0, 2, 9, 8, 8, 6, 6, 9, 4, 6, 6, 3

Offset: 1

Charles R Greathouse IV, Aug 31 2016

Let F(x) be the number of Fibonacci integers, A178772, less than or equal to x. Then exp(c*sqrt(log x) - (log x)^e) < F(x) < exp(c*sqrt(log x) + (log x)^(1/6 + e)) for any e > 0, where c is this constant. Luca, Pomerance, & Wagner conjecture that 1/6 can be replaced by 0, and note that it can be replaced by 1/8 on a strong form of the abc conjecture.

			5.1551243400746440551416193375652282874857604518811002483143110776973502988669...

Florian Luca, Carl Pomerance, Stephan Wagner, Fibonacci Integers, J. Number Theory 131 (2011), pp. 440-457. [conference version]

Cf. A178772.

RealDigits[2 Zeta[2] Sqrt[Zeta[3]/Zeta[6]/Log[GoldenRatio]], 10, 81][[1]] (* Indranil Ghosh, Mar 19 2017 *)

phi=(sqrt(5)+1)/2
2*zeta(2)*sqrt(zeta(3)/zeta(6)/log(phi))

2*zeta(2)*sqrt(zeta(3)/zeta(6)/log(phi)) where phi = (1 + sqrt(5))/2 is the golden ratio.

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A065108 Positive numbers expressible as a product of Fibonacci numbers.

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A178777 Numbers that are not Fibonacci integers.

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A201010 Integers that can be written as the product and/or quotient of Lucas numbers.

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A185060 Number of Fibonacci integers in the interval [1, 10^n].

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A275976 Decimal expansion of a constant relating to the density of Fibonacci integers.

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