A114566 Number of prime factors of A083216(n), counted with multiplicity.

4, 2, 4, 6, 3, 5, 4, 5, 2, 10, 5, 3, 3, 3, 4, 10, 5, 7, 2, 4, 5, 10, 4, 4, 2, 4, 5, 7, 3, 5, 5, 4, 3, 8, 4, 6, 4, 6, 4, 7, 5, 4, 3, 3, 4, 10, 5, 6, 4, 5, 5, 7, 3, 5, 6, 6, 4, 10, 5, 6, 7, 4, 4, 7, 5, 9, 4, 4, 5, 8, 2, 6, 6, 5, 5, 6, 4, 5, 5, 7, 3, 7, 5, 4, 6

Offset: 0

Author

Jonathan Vos Post, Feb 15 2006

Keywords

• nonn

Examples

			a(0) = 4 because Wilf(0) = 20615674205555510 = 2 * 5 * 5623 * 366631232537 has 4 prime factors with multiplicity.
a(1) = 2 because Wilf(1) is semiprime, namely 3794765361567513 = 3 * 1264921787189171.
a(2) = 4 because Wilf(2) = 24410439567123023 = 823 * 1069 * 5779 * 4801151.
a(3) = 6 because Wilf(3) = 2^3 * 1039 * 4481 * 757266563 (note that the prime factor 2 is counted 3 times).
a(4) = 3 because Wilf(4) = 52615644495813559 = 983 * 2521 * 21231883913.
a(5) = 5 because Wilf(5) = 80820849424504095 = 3^2 * 5 * 43 * 41767880839537.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..336
• R. L. Graham, A Fibonacci-Like Sequence of Composite Numbers, Math. Mag. 37, 1964, pp. 322-324.
• D. E. Knuth, A Fibonacci-Like Sequence of Composite Numbers, Math. Mag. 63, 1990, pp. 21-25.
• H. S. Wilf, Letter to the Editor, Math. Mag. 63, 1990, p. 284.
• Eric Weisstein's World of Mathematics, Primefree Sequence.

Crossrefs

Cf. A001222, A083216.

Programs

Maple

a:= n-> numtheory[bigomega]((<<0|1>, <1|1>>^n.
    <<20615674205555510, 3794765361567513>>)[1, 1]):
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 20 2017

Mathematica

PrimeOmega[LinearRecurrence[{1,1},{20615674205555510,3794765361567513},100]] (* Paolo Xausa, Nov 07 2023 *)

PARI

A083216(n)=if(n==0, 20615674205555510, if(n==1, 3794765361567513, A083216(n-1)+A083216(n-2)));
A114566(n)=bigomega(A083216(n));
for(n=0,30, print1(A114566(n),", ")) \\ R. J. Mathar, Dec 05 2007

Formula

a(n) = Omega(A083216(n)) = A001222(A083216(n)).

a(n) > 1 for all n >= 0.

Extensions

Corrected and extended by R. J. Mathar, Dec 05 2007

More terms from Alois P. Heinz, Sep 20 2017

Name edited by Michel Marcus, Nov 07 2023