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 1-4 of 4 results.

A303217 A(n,k) is the n-th index of a Fibonacci number with exactly k distinct prime factors; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

3, 8, 4, 15, 9, 5, 20, 16, 10, 6, 30, 24, 18, 12, 7, 40, 36, 27, 21, 14, 11, 70, 48, 42, 28, 33, 19, 13, 60, 81, 54, 44, 32, 35, 22, 17, 80, 72, 104, 56, 45, 52, 37, 25, 23, 90, 84, 110, 105, 64, 50, 55, 38, 26, 29, 140, 126, 88, 112, 136, 78, 57, 74, 39, 31, 43
Offset: 1

Views

Author

Alois P. Heinz, Apr 19 2018

Keywords

Examples

			Square array A(n,k) begins:
   3,  8, 15, 20, 30,  40,  70,  60,  80,  90, ...
   4,  9, 16, 24, 36,  48,  81,  72,  84, 126, ...
   5, 10, 18, 27, 42,  54, 104, 110,  88, 165, ...
   6, 12, 21, 28, 44,  56, 105, 112,  96, 256, ...
   7, 14, 33, 32, 45,  64, 136, 114, 100, 258, ...
  11, 19, 35, 52, 50,  78, 148, 128, 108, 266, ...
  13, 22, 37, 55, 57,  92, 152, 130, 132, 296, ...
  17, 25, 38, 74, 63,  95, 164, 135, 138, 304, ...
  23, 26, 39, 77, 66,  99, 182, 147, 156, 322, ...
  29, 31, 46, 85, 68, 102, 186, 154, 184, 369, ...

Links

Crossrefs

Columns k=2-16 give: A114842, A114841, A114843, A114840, A114839, A114838, A114837, A114836, A114826, A114825, A114824, A114823, A117529, A117551, A117550.
Row n=1 gives: A060320.
Cf. A000045, A001221, A022307, A303215, A303218.

Programs

  • Maple
    F:= combinat[fibonacci]: with(numtheory):
A:= proc() local h, p, q; p, q:= proc() [] end, 2;
      proc(n, k)
        while nops(p(k))
  • Mathematica
    nmax = 12; maxIndex = 200;
nu[n_] := nu[n] = PrimeNu[Fibonacci[n]];
col[k_] := Select[Range[maxIndex], nu[#] == k&];
T = Array[col, nmax];
A[n_, k_] := T[[k, n]];
Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 04 2020 *)

Formula

A000045(A(n,k)) = A303218(n,k).
A001221(A000045(A(n,k))) = k.

A060319 Smallest Fibonacci number with n distinct prime factors.

Original entry on oeis.org

1, 2, 21, 610, 6765, 832040, 102334155, 190392490709135, 1548008755920, 23416728348467685, 2880067194370816120, 81055900096023504197206408605, 2706074082469569338358691163510069157, 5358359254990966640871840, 57602132235424755886206198685365216, 18547707689471986212190138521399707760
Offset: 0

Views

Author

Labos Elemer, Mar 28 2001

Keywords

Examples

			a(5) = F(30) = 832040 = 2^3 * 5 * 11 * 41 * 61.

Links

Crossrefs

Cf. A001605, A005478, A060320.
Row n=1 of A303218.

Programs

  • Mathematica
    f[n_]:=Length@FactorInteger[Fibonacci[n]]; lst={};Do[Do[If[f[n]==q,Print[Fibonacci[n]];AppendTo[lst,Fibonacci[n]];Break[]],{n,280}],{q,18}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 23 2009 *)
First /@ SortBy[#, Last] &@ Map[#[[1]] &, Values@ GroupBy[#, Last]] &@ Table[{#, PrimeNu@ #} &@ Fibonacci@ n, {n, 2, 200}] (* Michael De Vlieger, Feb 18 2017, Version 10 *)

A303216 A(n,k) is the n-th Fibonacci number with exactly k prime factors (counted with multiplicity); square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

2, 21, 3, 8, 34, 5, 6765, 610, 55, 13, 2584, 196418, 987, 377, 89, 144, 701408733, 317811, 10946, 4181, 233, 832040, 102334155, 1134903170, 2178309, 75025, 17711, 1597, 86267571272, 267914296, 12586269025, 365435296162, 32951280099, 3524578, 121393, 28657
Offset: 1

Views

Author

Alois P. Heinz, Apr 19 2018

Keywords

Examples

			Square array A(n,k) begins:
    2,    21,       8,         6765,           2584,                 144, ...
    3,    34,     610,       196418,      701408733,           102334155, ...
    5,    55,     987,       317811,     1134903170,         12586269025, ...
   13,   377,   10946,      2178309,   365435296162,      10610209857723, ...
   89,  4181,   75025,  32951280099,  6557470319842,    2111485077978050, ...
  233, 17711, 3524578, 139583862445, 72723460248141, 7540113804746346429, ...

Links

Crossrefs

Columns k=1-2 give: A005478, A053409.
Row n=1 gives A072397.
Cf. A000045, A001222, A303215, A303218.

Programs

  • Maple
    F:= combinat[fibonacci]: with(numtheory):
A:= proc() local h, p, q; p, q:= proc() [] end, 2;
      proc(n, k)
        while nops(p(k))
  • Mathematica
    A[n_, k_] := Module[{F = Fibonacci, h, p, q = 2}, p[_] = {}; While[ Length[p[k]] < n, q = q+1; h = PrimeOmega[F[q]]; p[h] = Append[p[h], F[q]]]; p[k][[n]]];
Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 10}] // Flatten (* Jean-François Alcover, Feb 05 2021, after Alois P. Heinz *)

Formula

A(n,k) = A000045(A303215(n,k)).
A001222(A(n,k)) = k.

A137563 Fibonacci numbers with three distinct prime divisors.

Original entry on oeis.org

610, 987, 2584, 10946, 3524578, 9227465, 24157817, 39088169, 63245986, 1836311903, 7778742049, 20365011074, 591286729879, 4052739537881, 17167680177565, 44945570212853, 61305790721611591, 420196140727489673, 1500520536206896083277, 6356306993006846248183
Offset: 1

Views

Author

Parthasarathy Nambi, Apr 25 2008

Keywords

Examples

			The distinct prime divisors of the Fibonacci number 610 are 2, 5 and 61.
The distinct prime divisors of the Fibonacci number 44945570212853 are 269, 116849 and 1429913.

Links

Crossrefs

Cf. A053409, A114841.
Intersection of A033992 and A000045. - Michel Marcus, Mar 24 2018
Column k=3 of A303218.

Programs

  • GAP
    P1:=List([1..110],n->Fibonacci(n));;
P2:=List([1..Length(P1)],i->Filtered(DivisorsInt(P1[i]),IsPrime));;
a:=List(Filtered([1..Length(P2)],i->Length(P2[i])=3),j->P1[j]); # Muniru A Asiru, Mar 25 2018
  • Maple
    with(numtheory): with(combinat): a:=proc(n) if nops(factorset(fibonacci(n)))= 3 then fibonacci(n) else end if end proc: seq(a(n),n=1..110); # Emeric Deutsch, May 18 2008
  • Mathematica
    Select[Array[Fibonacci, 120], PrimeNu@ # == 3 &] (* Michael De Vlieger, Apr 10 2018 *)
  • PARI
    lista(nn) = for (n=1, nn, if (omega(f=fibonacci(n))==3, print1(f, ", "))); \\ Michel Marcus, Mar 24 2018

Formula

a(n) = A000045(A114841(n)). - Michel Marcus, Mar 24 2018

Extensions

More terms from Emeric Deutsch, May 18 2008
Showing 1-4 of 4 results.

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