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.

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

1,1

Links

Alois P. Heinz, Antidiagonals n = 1..16, flattened

Formula

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

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

Example

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, ...

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:= bigomega(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..10);

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 *)

Crossrefs

Columns k=1-2 give: A005478, A053409.

Row n=1 gives A072397.

Cf. A000045, A001222, A303215, A303218.

Sequence in context: A249646 A143247 A342079 * A331460 A303218 A162536

Adjacent sequences: A303213 A303214 A303215 * A303217 A303218 A303219

Keyword

nonn,tabl

Author

Alois P. Heinz, Apr 19 2018

Status

approved