This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A240606 #25 Apr 07 2025 13:41:59 %S A240606 0,0,2,0,3,1,0,0,2,4,0,3,0,0,2,0,1,1,0,2,0,0,1,5,0,0,6,0,1,3,0,0,1,1, %T A240606 0,3,0,0,4,2,0,0,1,0,2,2,0,5,0,0,1,0,2,1,0,0,1,1,0,3,0,0,1,0,6,1,0,2, %U A240606 0,0,4,5,0,0,2 %N A240606 Let the prime factorization of (2*n)! be 2^e_1*3^e_2*5^e_3*...; then a(n) = maximal k such that e_1, ..., e_k are all even. %C A240606 See comment in A240537. According to Berend's theorem, the sequence is unbounded. %D A240606 P. Erdős, P. L. Graham, Old and new problems and results in combinatorial number theory, L'Enseignement Mathematique, Imprimerie Kunding, Geneva, 1980. %H A240606 Charles R Greathouse IV, <a href="/A240606/b240606.txt">Table of n, a(n) for n = 1..10000</a> %H A240606 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 A240606 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. %e A240606 (2*10)! = 2432902008176640000 = 2^18 * 3^8 * 5^4 * 7^2 * 11 * 13 * 17 * 19, and the first 4 exponents are even, so a(10) = 4. %t A240606 Map[Count[First[Split[Mod[Last[Transpose[FactorInteger[(2*#)!]]],2]]],0]&,Range[75]] (* _Peter J. C. Moses_, Apr 09 2014 *) %o A240606 (Sage) def a(n): %o A240606 f = list(factor(factorial(2*n))) %o A240606 c = -1 %o A240606 for pf in f: %o A240606 c = c + 1 %o A240606 if pf[1] % 2: %o A240606 return c # _Ralf Stephan_, Apr 09 2014 %o A240606 (PARI) fv(n,p)=my(s); while(n\=p, s+=n); s %o A240606 a(n)=n*=2; my(s); forprime(p=2,,if(fv(n,p)%2, return(s), s++)) \\ _Charles R Greathouse IV_, Apr 09 2014 %Y A240606 Cf. A240537. %K A240606 nonn %O A240606 1,3 %A A240606 _Vladimir Shevelev_, Apr 09 2014 %E A240606 More terms and example from _Ralf Stephan_, Apr 09 2014