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

Author

Alois P. Heinz, Apr 19 2018

Keywords

• nonn
• tabl

Examples

			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, ...
		

Links

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

Crossrefs

Column k=3 gives A137563.

Row n=1 gives: A060319.

Cf. A000045, A001221, A303216, A303217.

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(*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 *)

Formula

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

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