cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

6, 10, 12, 15, 16, 21, 24, 26, 28, 33, 39, 40, 42, 44, 54, 63, 65, 68, 72, 77, 87, 102, 104, 105, 110, 116, 126, 141, 165, 168, 170, 178, 188, 198, 203, 228, 267, 272, 273, 275, 288, 304, 319, 329, 369, 432, 440, 442, 445, 466, 492, 517, 522, 532, 597, 699
Offset: 1

Views

Author

Clark Kimberling, Jun 22 2016

Keywords

Comments

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.
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.
Guide to related sequences:
A274426 = union of (U = {F(i)F(j), 2 < i < j} and V = {L(i)L(j), 1 < i < j})
A274429 = union of (U = {F(i)F(j), 2 < i < j} and V = {L(i)L(j), 0 < i < j})
A274374 = union of (U = {F(i)F(j), 1 < i < j} and V = {L(i)L(j), 0 < i < j})

Examples

			U = {6,10,15,16,...}, V = {12,21,28,...}, so that A274426 = (6,10,12,15,16,21,...).

Links

Crossrefs

Cf. A274427 (positions of numbers in U), A274428 (positions of numbers in V), A000032, A000045, A274429, A274432.

Programs

  • Mathematica
    z = 200; f[n_] := Fibonacci[n];
u = Take[Sort[Flatten[Table[f[m] f[n], {n, 3, z}, {m, 3, n - 1}]]], z]
g[n_] := LucasL[n];
v = Take[Sort[Flatten[Table[g[u] g[v], {u, 2, z}, {v, 2, u - 1}]]], z]
Intersection[u, v] (* empty *)
w = Union[u, v]  (* A274426 *)
f1 = Select[Range[300], MemberQ[u, w[[#]]] &]  (* A274427 *)
g1 = Select[Range[300], MemberQ[v, w[[#]]] &]  (* A274428 *)

A274433 Products of two distinct tribonacci numbers > 1.

Original entry on oeis.org

15, 27, 45, 51, 85, 93, 153, 155, 171, 279, 285, 315, 513, 525, 527, 579, 945, 965, 969, 1065, 1737, 1767, 1775, 1785, 1959, 3195, 3255, 3265, 3281, 3603, 5877, 5983, 5985, 6005, 6035, 6627, 10809, 11001, 11005, 11045, 11101, 12189, 19881, 20235, 20243
Offset: 1

Views

Author

Clark Kimberling, Jun 22 2016

Keywords

Comments

Are these unique among all products of distinct tribonacci numbers (A000213)? (See A274432.)

Examples

			The tribonacci numbers > 1 are 3,5,9,17,31,57,..., so that the binary products in increasing order are 15, 27,45, 51, 85, ...

Crossrefs

Cf. A000213, A274432, A274434.

Programs

  • Mathematica
    r[1] := 1; r[2] := 1; r[3] = 1; r[n_] := r[n] = r[n - 1] + r[n - 2] + r[n - 3];
s = {1}; z = 60; f = Map[r, Range[z]]; Take[f, 20]
Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}];
Take[s, 2 z]  (* A274432 *)
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, 300}];
Map[Apply[Times, #] &, Select[ans, Length[#] == 2 &]] (* A274433 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 3 &]] (* A274434 *)
(* Peter J. C. Moses, Jun 17 2016 *)

A274434 Products of three distinct tribonacci numbers > 1.

Original entry on oeis.org

135, 255, 459, 465, 765, 837, 855, 1395, 1539, 1575, 1581, 2565, 2635, 2835, 2895, 2907, 4725, 4743, 4845, 5211, 5301, 5325, 5355, 8685, 8721, 8835, 8925, 9585, 9765, 9795, 9843, 15903, 15975, 16065, 16275, 16405, 17631, 17949, 17955, 18015, 18105, 29295
Offset: 1

Views

Author

Clark Kimberling, Jun 22 2016

Keywords

Comments

Are these unique among all products of distinct tribonacci numbers (A000213)? (See A274432.)

Examples

			The tribonacci numbers > 1 are 3,5,9,17,31,57,..., so that the trinary products in increasing order are 135, 255, 459, 465, 765,...

Crossrefs

Cf. A000213, A274432, A274433.

Programs

  • Mathematica
    r[1] := 1; r[2] := 1; r[3] = 1; r[n_] := r[n] = r[n - 1] + r[n - 2] + r[n - 3];
s = {1}; z = 60; f = Map[r, Range[z]]; Take[f, 20]
Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}];
Take[s, 2 z]  (* A274432 *)
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, 300}];
Map[Apply[Times, #] &, Select[ans, Length[#] == 2 &]] (* A274433 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 3 &]] (* A274434 *)
(* Peter J. C. Moses, Jun 17 2016 *)

A274452 Products of distinct Narayana's cow numbers (A000930).

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 12, 13, 18, 19, 24, 26, 27, 28, 36, 38, 39, 41, 48, 52, 54, 56, 57, 60, 72, 76, 78, 82, 84, 88, 104, 108, 112, 114, 117, 120, 123, 129, 144, 152, 156, 162, 164, 168, 171, 176, 180, 189, 216, 224, 228, 234, 240, 246, 247, 252, 258, 264, 277
Offset: 1

Views

Author

Clark Kimberling, Jun 23 2016

Keywords

Examples

			The Narayana's cow numbers numbers are 1, 2, 3, 4, 6, 9, 13, 19, 28, ..., so that the sequence of all products of distinct members, in increasing order, is (2, 3, 4, 6, 8, 9, 12, 13, 18, 19, 24, ...).

Links

Crossrefs

Cf. A160009, A274280, A274432.

Programs

  • Mathematica
    r[1] := 1; r[2] := 1; r[3] = 1; r[n_] := r[n] = r[n - 1] + r[n - 3]
s = {1}; z = 60; f = Map[r, Range[z]]; Take[f, 20] (*A000930*)
Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}];
Take[s, 2 z] (* A274452 *)

A274453 Products of distinct numbers in A052963.

Original entry on oeis.org

2, 5, 10, 14, 28, 40, 70, 80, 115, 140, 200, 230, 331, 400, 560, 575, 662, 953, 1120, 1150, 1610, 1655, 1906, 2744, 2800, 3220, 3310, 4600, 4634, 4765, 5488, 5600, 7901, 8050, 9200, 9268, 9530, 13240, 13342, 13720, 15802, 16100, 22750, 23000, 23170, 26480
Offset: 1

Views

Author

Clark Kimberling, Jun 23 2016

Keywords

Examples

			The numbers in A274453 are 1, 2, 5, 14, 40, 115, 331,..., so that the sequence of all products of distinct members, in increasing order, is (2, 5, 10, 14, 28, 40, 70, 80,...).

Links

Crossrefs

Cf. A160009, A274280, A274432, A274452.

Programs

  • Mathematica
    r[1] := 1; r[2] := 1; r[3] = 1; r[n_] := r[n] = 3 r[n - 1] - r[n - 3]
s = {1}; z = 30; f = Map[r, Range[z]]; Take[f, 20] (* A052963 *)
Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}];
Take[s, 2 z]  (* A274453 *)
Showing 1-5 of 5 results.

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