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A274354 Number of factors L(i) > 1 of A274281(n), where L = A000032 (Lucas numbers, 2,1,3,4,..., with 1 excluded).

%I A274354 #7 Jun 02 2025 12:19:04
%S A274354 1,1,1,2,1,2,1,2,2,1,2,2,3,2,1,2,2,3,2,1,2,3,2,3,2,1,2,3,2,3,2,3,2,1,
%T A274354 2,3,2,3,2,3,4,3,2,2,1,2,3,2,3,3,2,3,4,3,2,3,2,1,2,3,2,3,3,3,2,3,4,3,
%U A274354 4,2,3,2,1,2,2,3,3,2,3,3,4,3,2,3,4,3
%N A274354 Number of factors L(i) > 1 of A274281(n), where L = A000032 (Lucas numbers, 2,1,3,4,..., with 1 excluded).
%e A274354 The products of distinct Lucas numbers (including 2, excluding 1), arranged in increasing order, comprise A274281 (with 1 removed).  The list begins with 2, 3, 4, 6 = 2*3, 7, 8 = 2*4, 11, 12, 14, 18, 21, 22, 24 = 2*3*4, so that a(4) = 2, a(6) = 2, a(13) = 3.
%t A274354 r[1] := 2; r[2] := 1; r[n_] := r[n] = r[n - 1] + r[n - 2];
%t A274354 s = {1}; z = 40; f = Join[{2}, Map[r, 2 + Range[z]]]; Take[f, 10]
%t A274354 Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}];
%t A274354 infQ[n_] := MemberQ[f, n];
%t A274354 ans = Table[#[[Flatten[Position[Map[Apply[Times, #] &, #], s[[n]]]][[1]]]] &[
%t A274354 Rest[Subsets[Map[#[[1]] &, Select[Map[{#, infQ[#]} &,
%t A274354 Divisors[s[[n]]]], #[[2]] && #[[1]] > 1 &]]]]], {n, 2, 200}];
%t A274354 Take[ans, 10]
%t A274354 w = Map[Length, ans]
%t A274354 Map[Apply[Times, #] &, Select[ans, Length[#] == 2 &]] (* A274349 *)
%t A274354 Map[Apply[Times, #] &, Select[ans, Length[#] == 3 &]] (* A274350 *)
%t A274354 (* _Peter J. C. Moses_, Jun 17 2016 *)
%Y A274354 Cf. A000032, A274353, A274349, A274350, A160009, A272947.
%K A274354 nonn,easy
%O A274354 1,4
%A A274354 _Clark Kimberling_, Jun 18 2016