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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.

%I A303218 #23 Feb 05 2021 14:29:40
%S A303218 2,21,3,610,34,5,6765,987,55,8,832040,46368,2584,144,13,102334155,
%T A303218 14930352,196418,10946,377,89,190392490709135,4807526976,267914296,
%U A303218 317811,3524578,4181,233,1548008755920,37889062373143906,86267571272,701408733,2178309,9227465,17711,1597
%N 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.
%H A303218 Alois P. Heinz, <a href="/A303218/b303218.txt">Antidiagonals n = 1..18,  flattened</a>
%F A303218 A(n,k) = A000045(A303217(n,k)).
%F A303218 A001221(A(n,k)) = k.
%e A303218 Square array A(n,k) begins:
%e A303218    2,   21,     610,        6765,      832040,        102334155, ...
%e A303218    3,   34,     987,       46368,    14930352,       4807526976, ...
%e A303218    5,   55,    2584,      196418,   267914296,      86267571272, ...
%e A303218    8,  144,   10946,      317811,   701408733,     225851433717, ...
%e A303218   13,  377, 3524578,     2178309,  1134903170,   10610209857723, ...
%e A303218   89, 4181, 9227465, 32951280099, 12586269025, 8944394323791464, ...
%p A303218 F:= combinat[fibonacci]: with(numtheory):
%p A303218 A:= proc() local h, p, q; p, q:= proc() [] end, 2;
%p A303218       proc(n, k)
%p A303218         while nops(p(k))<n do q:= q+1;
%p A303218           h:= nops(factorset(F(q)));
%p A303218           p(h):= [p(h)[], F(q)]
%p A303218         od; p(k)[n]
%p A303218       end
%p A303218     end():
%p A303218 seq(seq(A(n, 1+d-n), n=1..d), d=1..12);
%t A303218 nmax = 12(*rows*);
%t A303218 maxIndex = 200; (* increase if message "part does not exist" *)
%t A303218 nu[n_] := nu[n] = PrimeNu[Fibonacci[n]];
%t A303218 col[k_] := Select[Range[maxIndex], nu[#] == k &];
%t A303218 T = Array[col, nmax];
%t A303218 A[n_, k_] := Fibonacci[T[[k, n]]];
%t A303218 Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Feb 05 2021 *)
%Y A303218 Column k=3 gives A137563.
%Y A303218 Row n=1 gives: A060319.
%Y A303218 Cf. A000045, A001221, A303216, A303217.
%K A303218 nonn,tabl
%O A303218 1,1
%A A303218 _Alois P. Heinz_, Apr 19 2018