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 1-6 of 6 results.

A049862 Products of two Fibonacci numbers with distinct indices.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 10, 13, 15, 16, 21, 24, 26, 34, 39, 40, 42, 55, 63, 65, 68, 89, 102, 104, 105, 110, 144, 165, 168, 170, 178, 233, 267, 272, 273, 275, 288, 377, 432, 440, 442, 445, 466, 610, 699, 712, 714, 715, 720, 754, 987, 1131
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

There are no duplicates except for the trivial cases 1*F(j)=1*F(j) and F(i)*F(j)=F(j)*F(i). - Robert Israel, May 11 2016
The number 1 is included because 1 = F(1)*F(2). - Clark Kimberling, Jun 19 2016

Links

Crossrefs

Cf. A000045, A160009, A272949.

Programs

  • Maple
    fib:= combinat:-fibonacci:
sort(convert(select(`<`,{0,seq(seq(fib(i)*fib(j),i=j+1..100),j=1..100)},fib(101)),list)); # Robert Israel, May 11 2016
  • Mathematica
    Take[Union[Flatten[Table[Fibonacci[i]*Fibonacci[j], {i, 0, 100}, {j, i + 1, 100}]]], 100] (* Clark Kimberling, May 11 2016 *)
  • PARI
    isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
isok(n) = {if ((n==0) || (n==1), return (1)); fordiv(n, d, if (d^2 < n, if (isfib(d) && isfib(n/d), return (1)););); return(0);} \\ Michel Marcus, May 27 2019
  • PARI
    lista(nn) = {my(out = List([0])); for (i=0, nn, for (j=i+1, nn, listput(out, fibonacci(i)*fibonacci(j)););); Vec(vecsort(select(x->(x < fibonacci(nn+1)), out), , 8));} \\ Michel Marcus, May 27 2019

Extensions

Name changed to conform with A272949 et al. by Clark Kimberling, Jun 18 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

Views

Author

Clark Kimberling, May 13 2016

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

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]

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

Views

Author

Clark Kimberling, May 13 2016

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Clark Kimberling, May 14 2016

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

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]
Showing 1-6 of 6 results.

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