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

%I A114566 #30 Feb 16 2025 08:32:59
%S A114566 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,
%T A114566 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,
%U A114566 4,4,5,8,2,6,6,5,5,6,4,5,5,7,3,7,5,4,6
%N A114566 Number of prime factors of A083216(n), counted with multiplicity.
%H A114566 Alois P. Heinz, <a href="/A114566/b114566.txt">Table of n, a(n) for n = 0..336</a>
%H A114566 R. L. Graham, <a href="http://www.jstor.org/stable/2689243">A Fibonacci-Like Sequence of Composite Numbers</a>, Math. Mag. 37, 1964, pp. 322-324.
%H A114566 D. E. Knuth, <a href="http://www.jstor.org/stable/2691504">A Fibonacci-Like Sequence of Composite Numbers</a>, Math. Mag. 63, 1990, pp. 21-25.
%H A114566 H. S. Wilf, <a href="http://www.jstor.org/stable/2690956">Letter to the Editor</a>, Math. Mag. 63, 1990, p. 284.
%H A114566 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimefreeSequence.html">Primefree Sequence.</a>
%F A114566 a(n) = Omega(A083216(n)) = A001222(A083216(n)).
%F A114566 a(n) > 1 for all n >= 0.
%e A114566 a(0) = 4 because Wilf(0) = 20615674205555510 = 2 * 5 * 5623 * 366631232537 has 4 prime factors with multiplicity.
%e A114566 a(1) = 2 because Wilf(1) is semiprime, namely 3794765361567513 = 3 * 1264921787189171.
%e A114566 a(2) = 4 because Wilf(2) = 24410439567123023 = 823 * 1069 * 5779 * 4801151.
%e A114566 a(3) = 6 because Wilf(3) = 2^3 * 1039 * 4481 * 757266563 (note that the prime factor 2 is counted 3 times).
%e A114566 a(4) = 3 because Wilf(4) = 52615644495813559 = 983 * 2521 * 21231883913.
%e A114566 a(5) = 5 because Wilf(5) = 80820849424504095 = 3^2 * 5 * 43 * 41767880839537.
%p A114566 a:= n-> numtheory[bigomega]((<<0|1>, <1|1>>^n.
%p A114566     <<20615674205555510, 3794765361567513>>)[1, 1]):
%p A114566 seq(a(n), n=0..80);  # _Alois P. Heinz_, Sep 20 2017
%t A114566 PrimeOmega[LinearRecurrence[{1,1},{20615674205555510,3794765361567513},100]] (* _Paolo Xausa_, Nov 07 2023 *)
%o A114566 (PARI) A083216(n)=if(n==0, 20615674205555510, if(n==1, 3794765361567513, A083216(n-1)+A083216(n-2)));
%o A114566 A114566(n)=bigomega(A083216(n));
%o A114566 for(n=0,30, print1(A114566(n),", ")) \\ _R. J. Mathar_, Dec 05 2007
%Y A114566 Cf. A001222, A083216.
%K A114566 nonn
%O A114566 0,1
%A A114566 _Jonathan Vos Post_, Feb 15 2006
%E A114566 Corrected and extended by _R. J. Mathar_, Dec 05 2007
%E A114566 More terms from _Alois P. Heinz_, Sep 20 2017
%E A114566 Name edited by _Michel Marcus_, Nov 07 2023