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.

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

Views

Author

Clark Kimberling, Jun 18 2016

Keywords

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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]

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

Views

Author

Clark Kimberling, Jun 18 2016

Keywords

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

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]

A274375 Products of 2 distinct Fibonacci numbers and products of two distinct Lucas numbers (including 2), arranged in increasing order.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 18, 21, 22, 24, 26, 28, 29, 33, 34, 36, 39, 40, 42, 44, 47, 54, 55, 58, 63, 65, 68, 72, 76, 77, 87, 89, 94, 102, 104, 105, 110, 116, 123, 126, 141, 144, 152, 165, 168, 170, 178, 188, 198, 199, 203, 228, 233
Offset: 1

Views

Author

Clark Kimberling, Jun 19 2016

Keywords

Comments

Are 2,3,6,8,21 the only numbers that are a product of two distinct Fibonacci numbers and also a product of two distinct Lucas numbers (including 2)?

Links

Crossrefs

Cf. A049862, A274349, A274374.

Programs

  • Mathematica
    z = 400; f[n_] := Fibonacci[n];
s = Join[{0}, Take[Sort[Flatten[Table[f[m] f[n], {n, 2, z}, {m, 2, n - 1}]]], z]]
g[n_] := LucasL[n - 1]; t = Take[Sort[Flatten[Table[g[u] g[v], {u, 1, z}, {v, 1, u - 1}]]], z]
Union[s, t]

A274350 Products of three distinct Lucas numbers (2,3,4,7,11,18,...).

Original entry on oeis.org

24, 42, 56, 66, 84, 88, 108, 132, 144, 154, 174, 216, 231, 232, 252, 282, 308, 348, 376, 378, 396, 406, 456, 504, 564, 594, 608, 609, 638, 658, 738, 792, 812, 912, 957, 984, 987, 1034, 1044, 1064, 1194, 1276, 1316, 1386, 1476, 1551, 1566, 1592, 1596, 1672
Offset: 1

Views

Author

Clark Kimberling, Jun 18 2016

Keywords

Examples

			24 = 2*3*4, 88 = 2 * 4 * 11.

Crossrefs

Cf. A000032, A274348, A274349, A272949.

Programs

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

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jun 18 2016

Keywords

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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 *)
Showing 1-5 of 5 results.

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