A303218 A(n,k) is the n-th Fibonacci number with exactly k distinct prime factors; square array A(n,k), n>=1, k>=1, read by antidiagonals.

2, 21, 3, 610, 34, 5, 6765, 987, 55, 8, 832040, 46368, 2584, 144, 13, 102334155, 14930352, 196418, 10946, 377, 89, 190392490709135, 4807526976, 267914296, 317811, 3524578, 4181, 233, 1548008755920, 37889062373143906, 86267571272, 701408733, 2178309, 9227465, 17711, 1597

Offset

1,1

Links

Alois P. Heinz, Antidiagonals n = 1..18,  flattened

Formula

A(n,k) = A000045(A303217(n,k)).

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

Example

Square array A(n,k) begins:
   2,   21,     610,        6765,      832040,        102334155, ...
   3,   34,     987,       46368,    14930352,       4807526976, ...
   5,   55,    2584,      196418,   267914296,      86267571272, ...
   8,  144,   10946,      317811,   701408733,     225851433717, ...
  13,  377, 3524578,     2178309,  1134903170,   10610209857723, ...
  89, 4181, 9227465, 32951280099, 12586269025, 8944394323791464, ...

Maple

F:= combinat[fibonacci]: with(numtheory):
A:= proc() local h, p, q; p, q:= proc() [] end, 2;
      proc(n, k)
        while nops(p(k))<n do q:= q+1;
          h:= nops(factorset(F(q)));
          p(h):= [p(h)[], F(q)]
        od; p(k)[n]
      end
    end():
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);

Mathematica

nmax = 12(*rows*);
maxIndex = 200; (* increase if message "part does not exist" *)
nu[n_] := nu[n] = PrimeNu[Fibonacci[n]];
col[k_] := Select[Range[maxIndex], nu[#] == k &];
T = Array[col, nmax];
A[n_, k_] := Fibonacci[T[[k, n]]];
Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Feb 05 2021 *)

Crossrefs

Column k=3 gives A137563.

Row n=1 gives: A060319.

Cf. A000045, A001221, A303216, A303217.

Sequence in context: A342079 A303216 A331460 * A162536 A100980 A360980

Adjacent sequences: A303215 A303216 A303217 * A303219 A303220 A303221

Keyword

nonn,tabl

Author

Alois P. Heinz, Apr 19 2018

Status

approved