A160009 Numbers that are the product of distinct Fibonacci numbers.

0, 1, 2, 3, 5, 6, 8, 10, 13, 15, 16, 21, 24, 26, 30, 34, 39, 40, 42, 48, 55, 63, 65, 68, 78, 80, 89, 102, 104, 105, 110, 120, 126, 130, 144, 165, 168, 170, 178, 195, 204, 208, 210, 233, 240, 267, 272, 273, 275, 288, 312, 315, 330, 336, 340, 377, 390, 432, 440, 442, 445

Offset: 1

Author

T. D. Noe, Apr 29 2009

Keywords

• nonn

Comments

Starts the same as A049862, the product of two distinct Fibonacci numbers. This sequence has an infinite number of consecutive terms that are consecutive numbers (such as 15 and 16) because fib(k)*fib(k+3) and fib(k+1)*fib(k+2) differ by one for all k >= 0.

It follows from Carmichael's theorem that if u and v are finite sets of Fibonacci numbers such that (product of all the numbers in u) = (product of all the numbers in v), then u = v. The same holds for many other 2nd order linear recurrence sequences with constant coefficients. In the following guide to related "distinct product sequences", W = Wythoff array, A035513:

base sequence distinct-product sequence

A000045 (Fibonacci) A160009

A000032 (Lucas, without 2) A274280

A000032 (Lucas, with 2) A274281

A000285 (1,4,5,...) A274282

A022095 (1,5,6,...) A274283

A006355 (2,4,6,...) A274284

A013655 (2,5,7,...) A274285

A022086 (3,6,9,...) A274191

row 2 of W: (4,7,11,...) A274286

row 3 of W: (6,10,16,...) A274287

row 4 of W: (9,15,24,...) A274288

- Clark Kimberling, Jun 17 2016

Links

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

Crossrefs

A059844, A065108

Programs

Mathematica

s={1}; nn=30; f=Fibonacci[2+Range[nn]]; Do[s=Union[s,Select[s*f[[i]],#<=f[[nn]]&]], {i,nn}]; s=Prepend[s,0]