A274429 -id:A274429 - cp's OEIS frontend

A274430 Positions in A274429 of products of distinct Fibonacci numbers > 1.

3, 5, 8, 9, 12, 13, 17, 18, 19, 23, 24, 25, 30, 31, 32, 33, 38, 39, 40, 41, 47, 48, 49, 50, 51, 57, 58, 59, 60, 61, 68, 69, 70, 71, 72, 73, 80, 81, 82, 83, 84, 85, 93, 94, 95, 96, 97, 98, 99, 107, 108, 109, 110, 111, 112, 113, 122, 123, 124, 125, 126, 127

Offset: 1

Clark Kimberling, Jun 22 2016

Complement of A274431.

Cf. A274429, A274431.

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 *)

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

Original entry on oeis.org

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 *)

A274431 Positions in A274426 of products of distinct Lucas numbers > 1 (excluding 2).

Original entry on oeis.org

1, 2, 4, 6, 7, 10, 11, 14, 15, 16, 20, 21, 22, 26, 27, 28, 29, 34, 35, 36, 37, 42, 43, 44, 45, 46, 52, 53, 54, 55, 56, 62, 63, 64, 65, 66, 67, 74, 75, 76, 77, 78, 79, 86, 87, 88, 89, 90, 91, 92, 100, 101, 102, 103, 104, 105, 106, 114, 115, 116, 117, 118, 119

Offset: 1

Clark Kimberling, Jun 22 2016

Complement of A274430.

Cf. A274429, A274430.

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 *)

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A274430 Positions in A274429 of products of distinct Fibonacci numbers > 1.

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

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A274431 Positions in A274426 of products of distinct Lucas numbers > 1 (excluding 2).

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