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

0,1

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.

Formula

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

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A001222, A083216.

Sequence in context: A237869 A066978 A236187 * A013679 A096428 A091007

Adjacent sequences: A114563 A114564 A114565 * A114567 A114568 A114569

Keyword

nonn,changed

Author

Jonathan Vos Post, Feb 15 2006

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

Status

approved