cp's OEIS Frontend

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

%I A274426 #12 Oct 31 2017 06:56:11
%S A274426 6,10,12,15,16,21,24,26,28,33,39,40,42,44,54,63,65,68,72,77,87,102,
%T A274426 104,105,110,116,126,141,165,168,170,178,188,198,203,228,267,272,273,
%U A274426 275,288,304,319,329,369,432,440,442,445,466,492,517,522,532,597,699
%N A274426 Numbers that are a product of two distinct Fibonacci numbers >1 or two distinct Lucas numbers > 1.
%C A274426 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.
%C 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.
%C A274426 Guide to related sequences:
%C A274426 A274426 = union of (U = {F(i)F(j), 2 < i < j} and V = {L(i)L(j), 1 < i < j})
%C A274426 A274429 = union of (U = {F(i)F(j), 2 < i < j} and V = {L(i)L(j), 0 < i < j})
%C A274426 A274374 = union of (U = {F(i)F(j), 1 < i < j} and V = {L(i)L(j), 0 < i < j})
%H A274426 G. C. Greubel, <a href="/A274426/b274426.txt">Table of n, a(n) for n = 1..10000</a>
%e A274426 U = {6,10,15,16,...}, V = {12,21,28,...}, so that A274426 = (6,10,12,15,16,21,...).
%t A274426 z = 200; f[n_] := Fibonacci[n];
%t A274426 u = Take[Sort[Flatten[Table[f[m] f[n], {n, 3, z}, {m, 3, n - 1}]]], z]
%t A274426 g[n_] := LucasL[n];
%t A274426 v = Take[Sort[Flatten[Table[g[u] g[v], {u, 2, z}, {v, 2, u - 1}]]], z]
%t A274426 Intersection[u, v] (* empty *)
%t A274426 w = Union[u, v]  (* A274426 *)
%t A274426 f1 = Select[Range[300], MemberQ[u, w[[#]]] &]  (* A274427 *)
%t A274426 g1 = Select[Range[300], MemberQ[v, w[[#]]] &]  (* A274428 *)
%Y A274426 Cf. A274427 (positions of numbers in U), A274428 (positions of numbers in V), A000032, A000045, A274429, A274432.
%K A274426 nonn,easy
%O A274426 1,1
%A A274426 _Clark Kimberling_, Jun 22 2016