0,1

The number of prime divisors of n (counted with multiplicity) of the Wilf sequence. The Wilf sequence in question is A083216 "Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2)." This is a second-order linear recurrence sequence with a(0) and a(1) coprime that does not contain any primes. It was found by Herbert Wilf in 1990. That is, a(n) of this derived sequence is provably never 1. Is there a linear recurrence whose values include both primes and composites, but no semiprimes?

R. L. Graham, Math. Mag. 37, 1964, pp. 322-324.

D. E. Knuth, Math. Mag. 63, 1990, pp. 21-25.

H. S. Wilf, Letters to the Editor, Math. Mag. 63, 1990.

Eric Weisstein's World of Mathematics, Primefree Sequence.

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

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.

(PARI) A083216(n)={ if(n==0, return(20615674205555510), if(n==1, return(3794765361567513), return(A083216(n-1)+A083216(n-2)) ) ; ) ; } A114566(n)={ return(bigomega(A083216(n))) ; } { for(n=0, 30, print1(A114566(n), ", ") ; ) ; } - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 05 2007

Cf. A001222, A083216.

Sequence in context: A131953 A007005 A066978 this_sequence A013679 A096428 A091007

Adjacent sequences: A114563 A114564 A114565 this_sequence A114567 A114568 A114569

easy,nonn

Jonathan Vos Post (jvospost2(AT)yahoo.com), Feb 15 2006

Corrected and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 05 2007