A240668 - OEIS

%I #13 Apr 11 2014 02:31:44

%S 1,2,0,1,0,0,2,1,0,0,2,0,1,2,0,1,0,0,2,0,3,3,0,0,1,2,0,1,0,0,1,1,0,0,

%T 1,0,1,2,0,0,1,5,0,1,0,0,3,0,1,1,0,2,0,0,2,1,0,0,3,0,1,2,0,3,0,0,2,0,

%U 5,2,0,0,1,3,0,1,0,0,2,0,1,1,0,1,0,0,4

%N Number of the first odd exponents in the prime power factorization of (2*n)!.

%C According to Chen's theorem, the sequence is unbounded.

%H Peter J. C. Moses, <a href="/A240668/b240668.txt">Table of n, a(n) for n = 1..10000</a>

%H D. Berend, <a href="http://dx.doi.org/10.1006/jnth.1997.2106">Parity of exponents in the factorization of n!</a>, J. Number Theory, 64 (1997), 13-19.

%H Y.-G. Chen, <a href="http://dx.doi.org/10.1016/S0022-314X(03)00013-1">On the parity of exponents in the standard factorization of n!</a>, J. Number Theory, 100 (2003), 326-331.

%F a(n)*A240606(n) = 0.

%e 32! = 2^31*3^14*5^7*7^4*11^2*13^2*17*19*23*29*31, and only the first 1 exponent is odd, so a(16) = 1.

%t Map[Count[First[Split[Mod[Last[Transpose[FactorInteger[(2*#)!]]],2]]],1]&,Range[75]] (* _Peter J. C. Moses_, Apr 10 2014 *)

%o (PARI) a(n) = {my(f = factor((2*n)!)); my(nb = 0); my(i = 1); while((i <= #f~) && (f[i, 2] % 2), nb++; i++;); nb;} \\ _Michel Marcus_, Apr 10 2014

%Y Cf. A240537, A240606, A240619, A240620.

%K nonn

%O 1,2

%A _Vladimir Shevelev_, Apr 10 2014

%E More terms from _Michel Marcus_, Apr 10 2014