A274281 -id:A274281 - cp's OEIS frontend

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

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, 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, 4, 2, 3, 2, 1, 2, 2, 3, 3, 2, 3, 3, 4, 3, 2, 3, 4, 3

Offset: 1

Clark Kimberling, Jun 18 2016

			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.

Cf. A000032, A274353, A274349, A274350, A160009, A272947.

r[1] := 2; r[2] := 1; r[n_] := r[n] = r[n - 1] + r[n - 2];
s = {1}; z = 40; f = Join[{2}, Map[r, 2 + Range[z]]]; Take[f, 10]
Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}];
infQ[n_] := MemberQ[f, n];
ans = Table[#[[Flatten[Position[Map[Apply[Times, #] &, #], s[[n]]]][[1]]]] &[
Rest[Subsets[Map[#[[1]] &, Select[Map[{#, infQ[#]} &,
Divisors[s[[n]]]], #[[2]] && #[[1]] > 1 &]]]]], {n, 2, 200}];
Take[ans, 10]
w = Map[Length, ans]
Map[Apply[Times, #] &, Select[ans, Length[#] == 2 &]] (* A274349 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 3 &]] (* A274350 *)
(* Peter J. C. Moses, Jun 17 2016 *)

A160009 Numbers that are the product of distinct Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 10, 13, 15, 16, 21, 24, 26, 30, 34, 39, 40, 42, 48, 55, 63, 65, 68, 78, 80, 89, 102, 104, 105, 110, 120, 126, 130, 144, 165, 168, 170, 178, 195, 204, 208, 210, 233, 240, 267, 272, 273, 275, 288, 312, 315, 330, 336, 340, 377, 390, 432, 440, 442, 445

Offset: 1

T. D. Noe, Apr 29 2009

nonn

Starts the same as A049862, the product of two distinct Fibonacci numbers. This sequence has an infinite number of consecutive terms that are consecutive numbers (such as 15 and 16) because fib(k)*fib(k+3) and fib(k+1)*fib(k+2) differ by one for all k >= 0.
It follows from Carmichael's theorem that if u and v are finite sets of Fibonacci numbers such that (product of all the numbers in u) = (product of all the numbers in v), then u = v.  The same holds for many other 2nd order linear recurrence sequences with constant coefficients.  In the following guide to related "distinct product sequences", W = Wythoff array, A035513:
base sequence              distinct-product sequence
A000045 (Fibonacci)             A160009
A000032 (Lucas, without 2)      A274280
A000032 (Lucas, with 2)         A274281
A000285 (1,4,5,...)             A274282
A022095 (1,5,6,...)             A274283
A006355 (2,4,6,...)             A274284
A013655 (2,5,7,...)             A274285
A022086 (3,6,9,...)             A274191
row 2 of W: (4,7,11,...)        A274286
row 3 of W: (6,10,16,...)       A274287
row 4 of W: (9,15,24,...)       A274288
- Clark Kimberling, Jun 17 2016

T. D. Noe, Table of n, a(n) for n=1..1000

A059844, A065108

s={1}; nn=30; f=Fibonacci[2+Range[nn]]; Do[s=Union[s,Select[s*f[[i]],#<=f[[nn]]&]], {i,nn}]; s=Prepend[s,0]

A274280 Numbers that are a product of distinct Lucas numbers (1,3,4,7,11,...)

Original entry on oeis.org

1, 3, 4, 7, 11, 12, 18, 21, 28, 29, 33, 44, 47, 54, 72, 76, 77, 84, 87, 116, 123, 126, 132, 141, 188, 198, 199, 203, 216, 228, 231, 304, 308, 319, 322, 329, 348, 369, 378, 492, 504, 517, 521, 522, 532, 564, 594, 597, 609, 792, 796, 812, 836, 843, 846, 861

Offset: 1

Clark Kimberling, Jun 17 2016

See the Comment on distinct-product sequences in A160009.

			The Lucas numbers are 1,3,4,7,11,18,29,..., so that the sequence of all products of distinct Lucas numbers, in increasing order, are 1, 3, 4, 7, 11, 12, 18, 21, 28, 29,...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000204, A160009, A274281 (includes 2).

f[1] = 1; f[2] = 3; z = 32; f[n_] := f[n - 1] + f[n - 2]; f = Table[f[n], {n, 1, z}]; f
s = {1}; Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}]; s
Take[Times@@@Subsets[LucasL[Range[20]]]//Union,60] (* Harvey P. Dale, Sep 26 2019 *)

cp's OEIS Frontend

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

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A160009 Numbers that are the product of distinct Fibonacci numbers.

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A274280 Numbers that are a product of distinct Lucas numbers (1,3,4,7,11,...)

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