A271354 -id:A271354 - cp's OEIS frontend

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A271356 Difference sequence of the sequence A271354 of the increasing sequence of products of two distinct Fibonacci numbers greater than 1.

4, 5, 1, 8, 2, 13, 1, 2, 21, 2, 3, 34, 2, 1, 5, 55, 3, 2, 8, 89, 5, 1, 2, 13, 144, 8, 2, 3, 21, 233, 13, 2, 1, 5, 34, 377, 21, 3, 2, 8, 55, 610, 34, 5, 1, 2, 13, 89, 987, 55, 8, 2, 3, 21, 144, 1597, 89, 13, 2, 1, 5, 34, 233, 2584, 144, 21, 3, 2, 8, 55, 377

Offset: 1

Clark Kimberling, May 02 2016

Conjecture: every term except the first is a Fibonacci number.

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

Cf. A271354, A000045, A049998.

z = 100; f[n_] := Fibonacci[n];
t = Take[Sort[Flatten[Table[f[m] f[n], {n, 3, z}, {m, 3, n - 1}]]], 1000];
Differences[t]

A272949 Products of three distinct Fibonacci numbers > 1.

Original entry on oeis.org

30, 48, 78, 80, 120, 126, 130, 195, 204, 208, 210, 312, 315, 330, 336, 340, 504, 510, 520, 534, 544, 546, 550, 816, 819, 825, 840, 864, 880, 884, 890, 1320, 1326, 1335, 1360, 1365, 1398, 1424, 1428, 1430, 1440, 2136, 2142, 2145, 2160, 2184, 2200, 2210, 2262

Offset: 1

Clark Kimberling, May 13 2016

			a(1) = 30 = 2*3*5.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000045, A160009, A272947, A271354, A273950.

s = {1}; nn = 60; f = Fibonacci[2 + Range[nn]]; Do[s = Union[s, Select[s*f[[i]], # <= f[[nn]] &]], {i, nn}]; s =  Prepend[s, 0]; Take[s, 100]  (* A160009 *)
isFibonacciQ[n_] := Apply[Or, Map[IntegerQ, Sqrt[{# + 4, # - 4} &[5 n^2]]]];
ans = Join[{{0}}, {{1}}, Table[#[[Flatten[Position[Map[Apply[Times, #] &, #], s[[n]]]][[1]]]] &[Rest[Subsets[Rest[Map[#[[1]] &, Select[Map[{#, isFibonacciQ[#]} &, Divisors[s[[n]]]], #[[2]] &]]]]]], {n, 3, 500}]]
Map[Length, ans] (* A272947 *)
Flatten[Position[Map[Length, ans], 1]]  (* A272948 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 1 &]]  (* A000045 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 2 &]]  (* A271354 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 3 &]]  (* A272949 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 4 &]]  (* A272950 *)
(* Peter J. C. Moses, May 11 2016 *)
up=10^9; F=Fibonacci; i=3; Union[ Reap[ While[(a = F[i++]) < up, j=i; While[ (b = F[j++]*a) < up, h=j; While[ (c = F[h++]*b) < up, Sow@c ]]]][[2, 1]]] (* Giovanni Resta, May 14 2016 *)

A272947 Number of factors Fibonacci(i) > 1 of A160009(n+1).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 13 2016

			A160009(15) = 30 = 2*3*5, so that a(15) = 3.

MathOverflow, Distinctness of products of Fibonacci numbers

Cf. A000045, A160009, A272948, A271354, A272949, A273950.

s = {1}; nn = 60; f = Fibonacci[2 + Range[nn]]; Do[s = Union[s, Select[s*f[[i]], # <= f[[nn]] &]], {i, nn}]; s =  Prepend[s, 0]; Take[s, 100]  (* A160009 *)
isFibonacciQ[n_] := Apply[Or, Map[IntegerQ, Sqrt[{# + 4, # - 4} &[5 n^2]]]];
ans = Join[{{0}}, {{1}}, Table[#[[Flatten[Position[Map[Apply[Times, #] &, #], s[[n]]]][[1]]]] &[Rest[Subsets[Rest[Map[#[[1]] &, Select[Map[{#, isFibonacciQ[#]} &, Divisors[s[[n]]]], #[[2]] &]]]]]], {n, 3, 500}]]
Map[Length, ans] (* A272947 *)
Flatten[Position[Map[Length, ans], 1]]  (* A272948 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 1 &]]  (* A000045 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 2 &]]  (* A271354 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 3 &]]  (* A272949 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 4 &]]  (* A272950 *)
(* Peter J. C. Moses, May 11 2016 *)

A274349 Products of two distinct Lucas numbers (2,3,4,7,11,18,...).

Original entry on oeis.org

6, 8, 12, 14, 21, 22, 28, 33, 36, 44, 54, 58, 72, 77, 87, 94, 116, 126, 141, 152, 188, 198, 203, 228, 246, 304, 319, 329, 369, 398, 492, 517, 522, 532, 597, 644, 796, 836, 846, 861, 966, 1042, 1288, 1353, 1363, 1368, 1393, 1563, 1686, 2084, 2189, 2204, 2214

Offset: 1

Clark Kimberling, Jun 18 2016

			6 = 2*3, 44 = 4*11.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000032, A274348, A274347, A271354.

N:= 10000: # for terms <= N
L:= gfun:-rectoproc({f(n)=f(n-1)+f(n-2),f(0)=2,f(1)=1},f(n),remember):
S:= {}:
for i from 2 do
  u:= L(i);
  if u > N then break fi;
  for j from 0 to i-1 do
    if j = 1 then next fi;
    v:= u*L(j);
    if v > N then break fi;
    S:= S union {v};
od od:
sort(convert(S,list)); # Robert Israel, Jan 01 2021

z = 100; f[n_] := LucasL[n]; f[1] = 2 ;
Take[Sort[Flatten[Table[f[u] f[v], {u, 1, z}, {v, 1, u - 1}]]], z]
Take[Times@@@Subsets[Join[{2},LucasL[Range[2,20]]],{2}]//Union,60] (* Harvey P. Dale, Aug 13 2019 *)

A274348 Products of three distinct Lucas numbers (3,4,7,11,18,...)

Original entry on oeis.org

84, 132, 216, 231, 308, 348, 378, 504, 564, 594, 609, 792, 812, 912, 957, 987, 1276, 1316, 1386, 1476, 1551, 1566, 1596, 2068, 2088, 2128, 2233, 2388, 2508, 2538, 2583, 3344, 3384, 3444, 3619, 3654, 3864, 4059, 4089, 4104, 4179, 5412, 5452, 5472, 5572, 5742

Offset: 1

Clark Kimberling, Jun 18 2016

			84 = 3*4*7, 132 = 3*4*11.

Cf. A000032, A274349, A274348, A271354, A272949.

z = 100; f[n_] := LucasL[n];
Take[Sort[Flatten[Table[f[u] f[v] f[w], {u, 2, z}, {v, 2, u - 1}, {w, 2, v - 1}]]], z]

A272948 Positions of Fibonacci numbers in ordered sequence A160009 of all products of Fibonacci numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 27, 35, 44, 56, 70, 87, 108, 133, 163, 199, 242, 292, 352, 421, 504, 599, 712, 841, 994, 1167, 1371, 1602, 1873, 2179, 2535, 2936, 3401, 3924, 4528, 5206, 5985, 6858, 7857, 8976, 10252, 11679, 13299, 15109, 17159, 19446, 22028

Offset: 1

Clark Kimberling, May 13 2016

nonn

			A160009 = (0,1,2,3,5,6,8,10,13,15,16,21,...), so that a = (1,2,3,4,5,7,9,12,...).

Rémy Sigrist, Table of n, a(n) for n = 1..121
Distinctness of Fibonacci products
Rémy Sigrist, PARI program for A272948

Cf. A000045, A160009, A272947, A271354, A272949, A273950.

s = {1}; nn = 60; f = Fibonacci[2 + Range[nn]]; Do[s = Union[s, Select[s*f[[i]], # <= f[[nn]] &]], {i, nn}]; s =  Prepend[s, 0]; Take[s, 100]  (* A160009 *)
isFibonacciQ[n_] := Apply[Or, Map[IntegerQ, Sqrt[{# + 4, # - 4} &[5 n^2]]]];
ans = Join[{{0}}, {{1}}, Table[#[[Flatten[Position[Map[Apply[Times, #] &, #], s[[n]]]][[1]]]] &[Rest[Subsets[Rest[Map[#[[1]] &, Select[Map[{#, isFibonacciQ[#]} &, Divisors[s[[n]]]], #[[2]] &]]]]]], {n, 3, 500}]]
Map[Length, ans] (* A272947 *)
Flatten[Position[Map[Length, ans], 1]]  (* A272948 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 1 &]]  (* A000045 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 2 &]]  (* A271354 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 3 &]]  (* A272949 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 4 &]]  (* A272950 *)
(* Peter J. C. Moses, May 11 2016 *)

More terms from Rémy Sigrist, Mar 17 2019

A272950 Products of four distinct Fibonacci numbers > 1.

Original entry on oeis.org

240, 390, 624, 630, 1008, 1020, 1040, 1560, 1632, 1638, 1650, 1680, 2520, 2640, 2652, 2670, 2720, 2730, 4080, 4095, 4272, 4284, 4290, 4320, 4368, 4400, 4420, 6552, 6600, 6630, 6912, 6930, 6942, 6990, 7072, 7120, 7140, 7150, 10608, 10680, 10710, 10725, 10920

Offset: 1

Clark Kimberling, May 14 2016

			a(1) = 240 = 2*3*5*8.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000045, A160009, A272947, A271354, A272949.

s = {1}; nn = 60; f = Fibonacci[2 + Range[nn]]; Do[s = Union[s, Select[s*f[[i]], # <= f[[nn]] &]], {i, nn}]; s =  Prepend[s, 0]; Take[s, 100]  (* A160009 *)
isFibonacciQ[n_] := Apply[Or, Map[IntegerQ, Sqrt[{# + 4, # - 4} &[5 n^2]]]];
ans = Join[{{0}}, {{1}}, Table[#[[Flatten[Position[Map[Apply[Times, #] &, #], s[[n]]]][[1]]]] &[Rest[Subsets[Rest[Map[#[[1]] &, Select[Map[{#, isFibonacciQ[#]} &, Divisors[s[[n]]]], #[[2]] &]]]]]], {n, 3, 500}]]
Map[Length, ans] (* A272947 *)
Flatten[Position[Map[Length, ans], 1]]  (* A272948 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 1 &]]  (* A000045 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 2 &]]  (* A271354 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 3 &]]  (* A272949 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 4 &]]  (* A272950 *)
(* Peter J. C. Moses, May 11 2016 *)
up=10^6; F=Fibonacci; i=3; Union[ Reap[ While[(a = F[i++]) < up, j=i; While[ (b = F[j++]*a) < up, h=j; While[(c = F[h++]*b) < up, k=h; While[ (d = F[k++]*c) < up, Sow@d ]]]]][[2, 1]]] (* Giovanni Resta, May 14 2016 *)

list(lim)=my(v=List(),F,best=5,t2,t3,t4,j,k,l); while(fibonacci(best++)<=30*lim,); F=vector(best,i,fibonacci(i)); for(i=6,best, j=4; while(j++Charles R Greathouse IV, May 14 2016

A274347 Products of two distinct Lucas numbers (3,4,7,11,18,...).

Original entry on oeis.org

12, 21, 28, 33, 44, 54, 72, 77, 87, 116, 126, 141, 188, 198, 203, 228, 304, 319, 329, 369, 492, 517, 522, 532, 597, 796, 836, 846, 861, 966, 1288, 1353, 1363, 1368, 1393, 1563, 2084, 2189, 2204, 2214, 2254, 2529, 3372, 3542, 3567, 3572, 3582, 3647, 4092

Offset: 1

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Author

Clark Kimberling, Jun 18 2016

Keywords

nonn
easy

Comments

L(i)*L(j) = L(i+j) + (-1)^i*L(j-i). - Robert Israel, Sep 02 2019

Examples

12 = 3*4, 21 = 3*7.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000032, A274348, A274349, A271354.

Programs

Maple

L:= gfun:-rectoproc({f(n+1)=f(n)+f(n-1),f(0)=2,f(1)=1},f(n),remember): Q:= proc(n) local j; op(sort([seq(L(n)+(-1)^j*L(n-2*j),j=2..(n-1)/2)])) end proc: map(Q, [$5..20]); # Robert Israel, Sep 02 2019

Mathematica

z = 100; f[n_] := LucasL[n]; Take[Sort[Flatten[Table[f[u] f[v], {u, 2, z}, {v, 2, u - 1}]]], z]

A049999 a(n) = smallest index k such that Fibonacci(k) = d(n), where d = A049998 (sequence of first differences of ordered products of Fibonacci numbers, i.e., of A049997, with no duplicates).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 3, 1, 5, 4, 1, 1, 6, 5, 1, 3, 7, 6, 1, 1, 4, 8, 7, 3, 1, 5, 9, 8, 4, 1, 1, 6, 10, 9, 5, 1, 3, 7, 11, 10, 6, 1, 1, 4, 8, 12, 11, 7, 3, 1, 5, 9, 13, 12, 8, 4, 1, 1, 6, 10, 14, 13, 9, 5, 1, 3, 7, 11, 15, 14, 10, 6, 1, 1, 4, 8, 12, 16, 15, 11

Offset: 1

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Author

Clark Kimberling

Keywords

nonn

Comments

"David W. Wilson conjectured (Dec 14 2005) that" sequence A049998 "consists only of Fibonacci numbers. Proofs were found by Franklin T. Adams-Watters and Don Reble, Dec 14 2005." - Petros Hadjicostas, Nov 08 2019 [This comment was copied from A049998, which includes Don Reble's proof of the conjecture.]

Examples

From _Petros Hadjicostas_, Nov 08 2019: (Start) A049998(1) = 1 = Fibonacci(1) = Fibonacci(2), so a(1) = min(1,2) = 1. A049998(7) = 2 = Fibonacci(3), so a(7) = 3. A049998(10) = 3 = Fibonacci(4), so a(10) = 4. A049998(13) = 5 = Fibonacci(5), so a(13) = 5. A049998(17) = 8 = Fibonacci(6), so a(17) = 6. (End)

Links

Clark Kimberling, Orderings of products of Fibonacci numbers, Fibonacci Quarterly 42:1 (2004), pp. 28-35. (Includes a proof of the conjecture proved in the Comments for sequence A049998.)

Crossrefs

Cf. A000045, A049997, A049998, A094563, A226857, A271354.

Formula

A000045(a(n)) = A049998(n) = A049997(n) - A049997(n-1) for n >= 1. - Petros Hadjicostas, Nov 08 2019

Extensions

Name edited by and more terms from Petros Hadjicostas, Nov 08 2019

Showing 1-9 of 9 results.

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A271356 Difference sequence of the sequence A271354 of the increasing sequence of products of two distinct Fibonacci numbers greater than 1.

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A272949 Products of three distinct Fibonacci numbers > 1.

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Mathematica

A272947 Number of factors Fibonacci(i) > 1 of A160009(n+1).

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Mathematica

A274349 Products of two distinct Lucas numbers (2,3,4,7,11,18,...).

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Maple

Mathematica

A274348 Products of three distinct Lucas numbers (3,4,7,11,18,...)

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Mathematica

A272948 Positions of Fibonacci numbers in ordered sequence A160009 of all products of Fibonacci numbers.

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Mathematica

Extensions

A272950 Products of four distinct Fibonacci numbers > 1.

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PARI

A274347 Products of two distinct Lucas numbers (3,4,7,11,18,...).

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Maple

Mathematica

A049999 a(n) = smallest index k such that Fibonacci(k) = d(n), where d = A049998 (sequence of first differences of ordered products of Fibonacci numbers, i.e., of A049997, with no duplicates).

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