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Number of distinct prime factors in central binomial coefficients C(n, floor(n/2)), the terms of A001405.
11

%I #22 Jul 22 2022 16:43:46

%S 0,1,1,2,2,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,8,8,8,

%T 9,9,9,9,10,10,10,10,10,10,9,9,10,10,10,10,10,10,10,10,12,12,13,13,12,

%U 12,12,12,12,13,14,14,14,14,14,14,14,14,14,14,15,15,14,14,15,15,15,15,16

%N Number of distinct prime factors in central binomial coefficients C(n, floor(n/2)), the terms of A001405.

%C Sequence is not monotonic. E.g., a(44)=10, a(45)=9 and a(46)=10. The number of prime factors of n! is pi(n), but these numbers are lower.

%C Prime factors are counted without multiplicity. - _Harvey P. Dale_, May 20 2012

%H T. D. Noe, <a href="/A034973/b034973.txt">Table of n, a(n) for n = 1..10000</a>

%e a(25) = omega(binomial(25,12)) = omega(5200300) = 6 because the prime factors are 2, 5, 7, 17, 19, 23.

%t Table[PrimeNu[Binomial[n,Floor[n/2]]],{n,90}] (* _Harvey P. Dale_, May 20 2012 *)

%o (PARI) a(n)=omega(binomial(n,n\2)) \\ _Charles R Greathouse IV_, Apr 29 2015

%Y Cf. A001405, A034974, A067434.

%K nonn,easy,nice

%O 1,4

%A _Labos Elemer_