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