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A303217 A(n,k) is the n-th index of a Fibonacci number with exactly k distinct prime factors; square array A(n,k), n>=1, k>=1, read by antidiagonals.

%I A303217 #23 Jan 04 2020 14:47:02
%S A303217 3,8,4,15,9,5,20,16,10,6,30,24,18,12,7,40,36,27,21,14,11,70,48,42,28,
%T A303217 33,19,13,60,81,54,44,32,35,22,17,80,72,104,56,45,52,37,25,23,90,84,
%U A303217 110,105,64,50,55,38,26,29,140,126,88,112,136,78,57,74,39,31,43
%N A303217 A(n,k) is the n-th index of a Fibonacci number with exactly k distinct prime factors; square array A(n,k), n>=1, k>=1, read by antidiagonals.
%H A303217 Alois P. Heinz, <a href="/A303217/b303217.txt">Antidiagonals n = 1..18, flattened</a>
%F A303217 A000045(A(n,k)) = A303218(n,k).
%F A303217 A001221(A000045(A(n,k))) = k.
%e A303217 Square array A(n,k) begins:
%e A303217    3,  8, 15, 20, 30,  40,  70,  60,  80,  90, ...
%e A303217    4,  9, 16, 24, 36,  48,  81,  72,  84, 126, ...
%e A303217    5, 10, 18, 27, 42,  54, 104, 110,  88, 165, ...
%e A303217    6, 12, 21, 28, 44,  56, 105, 112,  96, 256, ...
%e A303217    7, 14, 33, 32, 45,  64, 136, 114, 100, 258, ...
%e A303217   11, 19, 35, 52, 50,  78, 148, 128, 108, 266, ...
%e A303217   13, 22, 37, 55, 57,  92, 152, 130, 132, 296, ...
%e A303217   17, 25, 38, 74, 63,  95, 164, 135, 138, 304, ...
%e A303217   23, 26, 39, 77, 66,  99, 182, 147, 156, 322, ...
%e A303217   29, 31, 46, 85, 68, 102, 186, 154, 184, 369, ...
%p A303217 F:= combinat[fibonacci]: with(numtheory):
%p A303217 A:= proc() local h, p, q; p, q:= proc() [] end, 2;
%p A303217       proc(n, k)
%p A303217         while nops(p(k))<n do q:= q+1;
%p A303217           h:= nops(factorset(F(q)));
%p A303217           p(h):= [p(h)[], (q)]
%p A303217         od; p(k)[n]
%p A303217       end
%p A303217     end():
%p A303217 seq(seq(A(n, 1+d-n), n=1..d), d=1..12);
%t A303217 nmax = 12; maxIndex = 200;
%t A303217 nu[n_] := nu[n] = PrimeNu[Fibonacci[n]];
%t A303217 col[k_] := Select[Range[maxIndex], nu[#] == k&];
%t A303217 T = Array[col, nmax];
%t A303217 A[n_, k_] := T[[k, n]];
%t A303217 Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Jan 04 2020 *)
%Y A303217 Columns k=2-16 give: A114842, A114841, A114843, A114840, A114839, A114838, A114837, A114836, A114826, A114825, A114824, A114823, A117529, A117551, A117550.
%Y A303217 Row n=1 gives: A060320.
%Y A303217 Cf. A000045, A001221, A022307, A303215, A303218.
%K A303217 nonn,tabl
%O A303217 1,1
%A A303217 _Alois P. Heinz_, Apr 19 2018