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Number of distinct prime factors in binomial(2*n,n).
17

%I #30 Aug 19 2024 17:45:26

%S 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 12,12,13,14,14,14,14,14,15,14,15,15,16,16,16,17,17,17,18,18,18,18,18,

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

%C a(n) = A001221(A000984(n)) = length of n-th row in A226078. - _Reinhard Zumkeller_, May 25 2013

%H T. D. Noe, <a href="/A067434/b067434.txt">Table of n, a(n) for n=1..10000</a>

%F a(n) ~ kn/log n, with k = log 4. - _Charles R Greathouse IV_, May 25 2013

%p a := n -> nops(numtheory:-factorset(binomial(2*n,n))):

%p seq(a(n), n=1..76); # _Peter Luschny_, Oct 31 2015

%t Table[Length[FactorInteger[Binomial[2 n, n]]], {n, 100}] (* _T. D. Noe_, Aug 17 2011 *)

%o (Haskell)

%o a067434 = a001221 . a000984 -- _Reinhard Zumkeller_, May 25 2013

%o (PARI) a(n)=omega(binomial(2*n,n)) \\ _Charles R Greathouse IV_, May 25 2013

%o (PARI) valp(n,p)=my(s);while(n\=p,s+=n);s

%o 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 (Python)

%o from math import comb

%o from sympy import primenu

%o def A067434(n): return primenu(comb(n<<1,n)) # _Chai Wah Wu_, Aug 19 2024

%Y Cf. A193990, A193991 (number of prime factors <= n and > n).

%K easy,nonn

%O 1,2

%A _Benoit Cloitre_, Feb 23 2002