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A303215 A(n,k) is the n-th index of a Fibonacci number with exactly k prime factors (counted with multiplicity); square array A(n,k), n>=1, k>=1, read by antidiagonals.

%I A303215 #26 Jun 22 2023 17:10:30
%S A303215 3,8,4,6,9,5,20,15,10,7,18,27,16,14,11,12,44,28,21,19,13,30,40,45,32,
%T A303215 25,22,17,54,42,50,57,52,33,26,23,24,78,56,64,63,55,35,31,29,36,80,
%U A303215 102,66,75,68,74,37,34,43,138,100,88,128,70,92,69,77,38,41,47
%N A303215 A(n,k) is the n-th index of a Fibonacci number with exactly k prime factors (counted with multiplicity); square array A(n,k), n>=1, k>=1, read by antidiagonals.
%H A303215 Alois P. Heinz, <a href="/A303215/b303215.txt">Antidiagonals n = 1..16, flattened</a>
%F A303215 A000045(A(n,k)) = A303216(n,k).
%F A303215 A001222(A000045(A(n,k))) = k.
%e A303215 Square array A(n,k) begins:
%e A303215    3,  8,  6, 20, 18,  12,  30,  54,  24,  36, ...
%e A303215    4,  9, 15, 27, 44,  40,  42,  78,  80, 100, ...
%e A303215    5, 10, 16, 28, 45,  50,  56, 102,  88, 114, ...
%e A303215    7, 14, 21, 32, 57,  64,  66, 128, 110, 165, ...
%e A303215   11, 19, 25, 52, 63,  75,  70, 130, 112, 174, ...
%e A303215   13, 22, 33, 55, 68,  92,  81, 135, 184, 256, ...
%e A303215   17, 26, 35, 74, 69,  95, 104, 147, 186, 266, ...
%e A303215   23, 31, 37, 77, 76,  99, 105, 154, 189, 273, ...
%e A303215   29, 34, 38, 85, 91, 116, 136, 170, 196, 282, ...
%e A303215   43, 41, 39, 87, 98, 117, 148, 171, 225, 296, ...
%p A303215 F:= combinat[fibonacci]: with(numtheory):
%p A303215 A:= proc() local h, p, q; p, q:= proc() [] end, 2;
%p A303215       proc(n, k)
%p A303215         while nops(p(k))<n do q:= q+1;
%p A303215           h:= bigomega(F(q));
%p A303215           p(h):= [p(h)[], (q)]
%p A303215         od; p(k)[n]
%p A303215       end
%p A303215     end():
%p A303215 seq(seq(A(n, 1+d-n), n=1..d), d=1..12);
%t A303215 A[n_, k_] := Module[{h, p, q = 2}, p[k] = {}; While[Length[p[k]]<n, q++; h = PrimeOmega[Fibonacci[q]]; AppendTo[p[h], q]]; p[k][[n]] ];
%t A303215 Table[A[n, 1+d-n], {d, 1, 12}, {n, 1, d}] // Flatten (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y A303215 Columns k=1-13 give: A001605, A072381, A114812, A114813, A114814, A114815, A114816, A114817, A114818, A114819, A114820, A114821, A114822.
%Y A303215 Row n=1 gives A072396.
%Y A303215 Cf. A000045, A001222, A038575, A303216, A303217.
%K A303215 nonn,tabl
%O A303215 1,1
%A A303215 _Alois P. Heinz_, Apr 19 2018