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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.

%I A303216 #23 Feb 05 2021 11:27:51
%S A303216 2,21,3,8,34,5,6765,610,55,13,2584,196418,987,377,89,144,701408733,
%T A303216 317811,10946,4181,233,832040,102334155,1134903170,2178309,75025,
%U A303216 17711,1597,86267571272,267914296,12586269025,365435296162,32951280099,3524578,121393,28657
%N 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.
%H A303216 Alois P. Heinz, <a href="/A303216/b303216.txt">Antidiagonals n = 1..16, flattened</a>
%F A303216 A(n,k) = A000045(A303215(n,k)).
%F A303216 A001222(A(n,k)) = k.
%e A303216 Square array A(n,k) begins:
%e A303216     2,    21,       8,         6765,           2584,                 144, ...
%e A303216     3,    34,     610,       196418,      701408733,           102334155, ...
%e A303216     5,    55,     987,       317811,     1134903170,         12586269025, ...
%e A303216    13,   377,   10946,      2178309,   365435296162,      10610209857723, ...
%e A303216    89,  4181,   75025,  32951280099,  6557470319842,    2111485077978050, ...
%e A303216   233, 17711, 3524578, 139583862445, 72723460248141, 7540113804746346429, ...
%p A303216 F:= combinat[fibonacci]: with(numtheory):
%p A303216 A:= proc() local h, p, q; p, q:= proc() [] end, 2;
%p A303216       proc(n, k)
%p A303216         while nops(p(k))<n do q:= q+1;
%p A303216           h:= bigomega(F(q));
%p A303216           p(h):= [p(h)[], F(q)]
%p A303216         od; p(k)[n]
%p A303216       end
%p A303216     end():
%p A303216 seq(seq(A(n, 1+d-n), n=1..d), d=1..10);
%t A303216 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]]];
%t A303216 Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 10}] // Flatten (* _Jean-François Alcover_, Feb 05 2021, after _Alois P. Heinz_ *)
%Y A303216 Columns k=1-2 give: A005478, A053409.
%Y A303216 Row n=1 gives A072397.
%Y A303216 Cf. A000045, A001222, A303215, A303218.
%K A303216 nonn,tabl
%O A303216 1,1
%A A303216 _Alois P. Heinz_, Apr 19 2018