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A067434 Number of distinct prime factors in binomial(2*n,n).

%I A067434 #30 Aug 19 2024 17:45:26
%S A067434 1,2,2,3,3,4,4,5,5,5,6,6,6,6,7,8,8,9,9,10,10,10,9,10,10,10,10,12,13,
%T A067434 12,12,13,14,14,14,14,14,15,14,15,15,16,16,16,17,17,17,18,18,18,18,18,
%U A067434 19,20,19,19,19,20,20,21,21,21,21,22,22,23,24,23,23,23,23,24,24,24,25,25
%N A067434 Number of distinct prime factors in binomial(2*n,n).
%C A067434 a(n) = A001221(A000984(n)) = length of n-th row in A226078. - _Reinhard Zumkeller_, May 25 2013
%H A067434 T. D. Noe, <a href="/A067434/b067434.txt">Table of n, a(n) for n=1..10000</a>
%F A067434 a(n) ~ kn/log n, with k = log 4. - _Charles R Greathouse IV_, May 25 2013
%p A067434 a := n -> nops(numtheory:-factorset(binomial(2*n,n))):
%p A067434 seq(a(n), n=1..76); # _Peter Luschny_, Oct 31 2015
%t A067434 Table[Length[FactorInteger[Binomial[2 n, n]]], {n, 100}] (* _T. D. Noe_, Aug 17 2011 *)
%o A067434 (Haskell)
%o A067434 a067434 = a001221 . a000984 -- _Reinhard Zumkeller_, May 25 2013
%o A067434 (PARI) a(n)=omega(binomial(2*n,n)) \\ _Charles R Greathouse IV_, May 25 2013
%o A067434 (PARI) valp(n,p)=my(s);while(n\=p,s+=n);s
%o A067434 a(n)=my(s);forprime(p=2,2*n,if(valp(2*n,p)>2*valp(n,p),s++)); s \\ _Charles R Greathouse IV_, May 25 2013
%o A067434 (Python)
%o A067434 from math import comb
%o A067434 from sympy import primenu
%o A067434 def A067434(n): return primenu(comb(n<<1,n)) # _Chai Wah Wu_, Aug 19 2024
%Y A067434 Cf. A193990, A193991 (number of prime factors <= n and > n).
%K A067434 easy,nonn
%O A067434 1,2
%A A067434 _Benoit Cloitre_, Feb 23 2002