A240668 Number of the first odd exponents in the prime power factorization of (2*n)!.

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, 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, 5, 2, 0, 0, 1, 3, 0, 1, 0, 0, 2, 0, 1, 1, 0, 1, 0, 0, 4

Offset: 1

Author

Vladimir Shevelev, Apr 10 2014

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..10000
• D. Berend, Parity of exponents in the factorization of n!, J. Number Theory, 64 (1997), 13-19.
• Y.-G. Chen, On the parity of exponents in the standard factorization of n!, J. Number Theory, 100 (2003), 326-331.

Crossrefs

Cf. A240537, A240606, A240619, A240620.

Programs

Mathematica

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

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

Formula

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

Extensions

More terms from Michel Marcus, Apr 10 2014