login
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.
5
2, 21, 3, 610, 34, 5, 6765, 987, 55, 8, 832040, 46368, 2584, 144, 13, 102334155, 14930352, 196418, 10946, 377, 89, 190392490709135, 4807526976, 267914296, 317811, 3524578, 4181, 233, 1548008755920, 37889062373143906, 86267571272, 701408733, 2178309, 9227465, 17711, 1597
OFFSET
1,1
FORMULA
A(n,k) = A000045(A303217(n,k)).
A001221(A(n,k)) = k.
EXAMPLE
Square array A(n,k) begins:
2, 21, 610, 6765, 832040, 102334155, ...
3, 34, 987, 46368, 14930352, 4807526976, ...
5, 55, 2584, 196418, 267914296, 86267571272, ...
8, 144, 10946, 317811, 701408733, 225851433717, ...
13, 377, 3524578, 2178309, 1134903170, 10610209857723, ...
89, 4181, 9227465, 32951280099, 12586269025, 8944394323791464, ...
MAPLE
F:= combinat[fibonacci]: with(numtheory):
A:= proc() local h, p, q; p, q:= proc() [] end, 2;
proc(n, k)
while nops(p(k))<n do q:= q+1;
h:= nops(factorset(F(q)));
p(h):= [p(h)[], F(q)]
od; p(k)[n]
end
end():
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);
MATHEMATICA
nmax = 12(*rows*);
maxIndex = 200; (* increase if message "part does not exist" *)
nu[n_] := nu[n] = PrimeNu[Fibonacci[n]];
col[k_] := Select[Range[maxIndex], nu[#] == k &];
T = Array[col, nmax];
A[n_, k_] := Fibonacci[T[[k, n]]];
Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Feb 05 2021 *)
CROSSREFS
Column k=3 gives A137563.
Row n=1 gives: A060319.
Sequence in context: A342079 A303216 A331460 * A162536 A377413 A100980
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Apr 19 2018
STATUS
approved