OFFSET
0,5
COMMENTS
Characterization: Let e_{p}(m) denote the exponent of the prime p in the prime factorization of m and [.] denote the Iverson bracket, then
e_{p}(a(n)) = Sum_{k>=1} [floor(n/p^k) is even].
This implies, among other things, that no prime > floor(n/2) can divide a(n). The prime exponents e_{2}(a(2n)) give Guy Steele's sequence GS(5,3) A080100.
Asymptotics: log a(n) ~ n(1 - log 2). It is conjectured that log a(n) ~ n(1 - log 2) + O(n^{1/2+eps}) for all eps > 0.
LINKS
Peter Luschny, Table of n, a(n) for n = 0..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
FORMULA
a(n) = 2^(-n)*Product_{1<=k<=n} A014963(k)*(k/2)^((-1)^k).
MAPLE
MATHEMATICA
a[0] = 1; a[n_] := LCM @@ Range[n] / (n! / Floor[n/2]!^2); Table[a[n], {n, 0, 46}] (* Jean-François Alcover, Jul 23 2013 *)
PROG
(PARI) L=1; X(n)={ ispower(n, , &n); if(isprime(n), n, 1); }
Y(n)={ a=X(n); b=if(bitand(1, n), a, a*(n/2)^2); L=(b*L)/n; }
A180000_list(n)={ L=1; vector(n, m, Y(m)); } \\ for n>0
(Sage)
def Exp(m, n) :
s = 0; p = m; q = n//p
while q > 0 :
if is_even(q) :
s = s + 1
p = p * m
q = n//p
return s
def A180000(n) :
A = [1, 1, 1, 1, 2, 2, 3, 3, 12]
if n < 9 : return A[n]
R = []; r = isqrt(n)
P = Primes(); p = P.first()
while p <= n//2 :
if p <= r : R.append(p^Exp(p, n))
elif p <= n//3 :
if is_even(n//p) : R.append(p)
else : R.append(p)
p = P.next(p)
return mul(x for x in R)
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Aug 17 2010
STATUS
approved