A274374 -id:A274374 - cp's OEIS frontend

A274426 Numbers that are a product of two distinct Fibonacci numbers >1 or two distinct Lucas numbers > 1.

6, 10, 12, 15, 16, 21, 24, 26, 28, 33, 39, 40, 42, 44, 54, 63, 65, 68, 72, 77, 87, 102, 104, 105, 110, 116, 126, 141, 165, 168, 170, 178, 188, 198, 203, 228, 267, 272, 273, 275, 288, 304, 319, 329, 369, 432, 440, 442, 445, 466, 492, 517, 522, 532, 597, 699

Offset: 1

Clark Kimberling, Jun 22 2016

Let U = {F(i)F(j), 2 < i < j}, where F = A000045 (Fibonacci numbers), and V = {L(i)L(j), 1 < 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 A274426.
Writing u for a Fibonacci product and v for a Lucas product, the numbers in A274426 are represented by the infinite word uuvuuvuuvvuuuvvuuuvvv... This is the concatenation of uuv and the words (u^k)(v^(k-1))(u^k)(v^k) for k >= 2. Thus, there are runs of Lucas products of every finite length and runs of Fibonacci products of every finite length except 1.
Guide to related sequences:
A274426 = union of (U = {F(i)F(j), 2 < i < j} and V = {L(i)L(j), 1 < i < j})
A274429 = union of (U = {F(i)F(j), 2 < i < j} and V = {L(i)L(j), 0 < i < j})
A274374 = union of (U = {F(i)F(j), 1 < i < j} and V = {L(i)L(j), 0 < i < j})

			U = {6,10,15,16,...}, V = {12,21,28,...}, so that A274426 = (6,10,12,15,16,21,...).

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A274427 (positions of numbers in U), A274428 (positions of numbers in V), A000032, A000045, A274429, A274432.

z = 200; f[n_] := Fibonacci[n];
u = Take[Sort[Flatten[Table[f[m] f[n], {n, 3, z}, {m, 3, n - 1}]]], z]
g[n_] := LucasL[n];
v = Take[Sort[Flatten[Table[g[u] g[v], {u, 2, z}, {v, 2, u - 1}]]], z]
Intersection[u, v] (* empty *)
w = Union[u, v]  (* A274426 *)
f1 = Select[Range[300], MemberQ[u, w[[#]]] &]  (* A274427 *)
g1 = Select[Range[300], MemberQ[v, w[[#]]] &]  (* A274428 *)

A274375 Products of 2 distinct Fibonacci numbers and products of two distinct Lucas numbers (including 2), arranged in increasing order.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 18, 21, 22, 24, 26, 28, 29, 33, 34, 36, 39, 40, 42, 44, 47, 54, 55, 58, 63, 65, 68, 72, 76, 77, 87, 89, 94, 102, 104, 105, 110, 116, 123, 126, 141, 144, 152, 165, 168, 170, 178, 188, 198, 199, 203, 228, 233

Offset: 1

Clark Kimberling, Jun 19 2016

Are 2,3,6,8,21 the only numbers that are a product of two distinct Fibonacci numbers and also a product of two distinct Lucas numbers (including 2)?

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A049862, A274349, A274374.

z = 400; f[n_] := Fibonacci[n];
s = Join[{0}, Take[Sort[Flatten[Table[f[m] f[n], {n, 2, z}, {m, 2, n - 1}]]], z]]
g[n_] := LucasL[n - 1]; t = Take[Sort[Flatten[Table[g[u] g[v], {u, 1, z}, {v, 1, u - 1}]]], z]
Union[s, t]

cp's OEIS Frontend

A274426 Numbers that are a product of two distinct Fibonacci numbers >1 or two distinct Lucas numbers > 1.

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