A274429 - OEIS

%I #5 Jun 22 2016 23:16:03

%S 3,4,6,7,10,11,12,15,16,18,21,24,26,28,29,33,39,40,42,44,47,54,63,65,

%T 68,72,76,77,87,102,104,105,110,116,123,126,141,165,168,170,178,188,

%U 198,199,203,228,267,272,273,275,288,304,319,322,329,369,432,440

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

%C 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.)

%C 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.

%C See A274426 for a guide to related sequences.

%t z = 200; f[n_] := Fibonacci[n];

%t u = Take[Sort[Flatten[Table[f[m] f[n], {n, 3, z}, {m, 3, n - 1}]]], 100]

%t g[n_] := LucasL[n];

%t v = Take[Sort[Flatten[Table[g[u] g[v], {u, 1, z}, {v, 1, u - 1}]]], z]

%t Intersection[u, v]

%t w = Union[u, v]  (* A274429 *)

%t Select[Range[300], MemberQ[u, w[[#]]] &]  (* A274430 *)

%t Select[Range[300], MemberQ[v, w[[#]]] &]  (* A274431 *)

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

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Jun 22 2016