A274429 Numbers that are a product of two distinct Fibonacci numbers >1 or two distinct Lucas numbers > 0 (excluding 2).

3, 4, 6, 7, 10, 11, 12, 15, 16, 18, 21, 24, 26, 28, 29, 33, 39, 40, 42, 44, 47, 54, 63, 65, 68, 72, 76, 77, 87, 102, 104, 105, 110, 116, 123, 126, 141, 165, 168, 170, 178, 188, 198, 199, 203, 228, 267, 272, 273, 275, 288, 304, 319, 322, 329, 369, 432, 440

Offset: 1

Author

Clark Kimberling, Jun 22 2016

Keywords

• nonn
• easy

Comments

Let U = {F(i)F(j), 2 < i < j}, where F = A000045 (Fibonacci numbers), and V = {L(i)L(j), 0 < i < j}, where L = A000032 (Lucas numbers). The sets U and V are disjoint, and their union, arranged as a sequence in increasing order, is A274429. (Unlike A274426, here all the Lucas numbers except 1 are included.)

Writing u for a Fibonacci product and v for a Lucas product, the numbers in A274429 are represented by the infinite word vvuvuvvuuvvuuvvvuuuvvvuuuvvvv... This is the concatenation of v and the words (v^k)(u^k)(v^k)(u^k) for k >= 1. Thus, there are runs of Fibonacci products of every finite length and runs of Lucas products of every finite length.

See A274426 for a guide to related sequences.

Crossrefs

Cf. A274430 (positions of numbers in U), A274431 (positions of numbers in V), A000032, A000045, A274426.

Programs

Mathematica

z = 200; f[n_] := Fibonacci[n];
u = Take[Sort[Flatten[Table[f[m] f[n], {n, 3, z}, {m, 3, n - 1}]]], 100]
g[n_] := LucasL[n];
v = Take[Sort[Flatten[Table[g[u] g[v], {u, 1, z}, {v, 1, u - 1}]]], z]
Intersection[u, v]
w = Union[u, v]  (* A274429 *)
Select[Range[300], MemberQ[u, w[[#]]] &]  (* A274430 *)
Select[Range[300], MemberQ[v, w[[#]]] &]  (* A274431 *)