A137563 -id:A137563 - cp's OEIS frontend

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.

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

Alois P. Heinz, Apr 19 2018

			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, ...

Alois P. Heinz, Antidiagonals n = 1..18, flattened

Column k=3 gives A137563.
Row n=1 gives: A060319.
Cf. A000045, A001221, A303216, A303217.

F:= combinat[fibonacci]: with(numtheory):
A:= proc() local h, p, q; p, q:= proc() [] end, 2;
      proc(n, k)
        while nops(p(k))

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 *)

A(n,k) = A000045(A303217(n,k)).

A001221(A(n,k)) = k.

A206096 Fibonacci numbers with 4 distinct prime divisors, each of multiplicity 1.

Original entry on oeis.org

6765, 196418, 317811, 2178309, 32951280099, 139583862445, 1304969544928657, 5527939700884757, 259695496911122585, 679891637638612258, 12200160415121876738, 83621143489848422977, 483162952612010163284885, 22698374052006863956975682

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 03 2012

nonn

Intersection of A000045 and A046386. - Michel Marcus, Sep 11 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..36

Cf. A000045, A053409, A137563

filter:= proc(t)
local F;
F:= ifactors(t)[2];
map(f -> f[2],F) = [1,1,1,1];
end proc:
select(filter, [seq(combinat:-fibonacci(n),n=1..200)]); # Robert Israel, Sep 07 2014

Select[Fibonacci[Range[200]],Last/@FactorInteger[#]=={1,1,1,1}&]

Vec(select(k -> omega(k)==4 && bigomega(k)==4, vector(100, i, fibonacci(i)))) \\ Edward Jiang, Sep 11 2014

A301561 Sphenic Fibonacci numbers.

Original entry on oeis.org

610, 987, 10946, 3524578, 9227465, 24157817, 39088169, 63245986, 1836311903, 7778742049, 20365011074, 591286729879, 4052739537881, 17167680177565, 44945570212853, 61305790721611591, 420196140727489673, 1500520536206896083277, 6356306993006846248183

Offset: 1

Waldemar Puszkarz, Mar 23 2018

nonn

Intersection of A000045 and A007304. There are 28 sphenic numbers among the first 200 positive Fibonacci numbers.

			610 is a term since it is a Fibonacci number that is a product of 3 distinct primes, 610=2*5*61, which makes it a sphenic number.

Cf. A000045, A007304, A061305 (squarefree Fibonaccis), A137563 (Fibonaccis with 3 distinct primes).

Select[Fibonacci@Range[120], SquareFreeQ[#]&&PrimeNu[#]==3&]

for(n=1, 120, fn=fibonacci(n); issquarefree(fn)&&omega(fn)==3&&print1(fn ","))

A000045 INTERSECT A007304.

cp's OEIS Frontend

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.

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A206096 Fibonacci numbers with 4 distinct prime divisors, each of multiplicity 1.

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A301561 Sphenic Fibonacci numbers.

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Mathematica

PARI

Formula