A180000 a(n) = lcm{1,2,...,n} / swinging_factorial(n) = A003418(n) / A056040(n).

1, 1, 1, 1, 2, 2, 3, 3, 12, 4, 10, 10, 30, 30, 105, 7, 56, 56, 252, 252, 1260, 60, 330, 330, 1980, 396, 2574, 286, 2002, 2002, 15015, 15015, 240240, 7280, 61880, 1768, 15912, 15912, 151164, 3876, 38760, 38760, 406980, 406980, 4476780, 99484, 1144066

Offset: 0

Author

Peter Luschny, Aug 17 2010

Keywords

• nonn

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.

Bounds: A056040(floor(n/3)) <= a(n) <= A056040(floor(n/2)) if n >= 285.

Links

• Peter Luschny, Table of n, a(n) for n = 0..1000
• Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Crossrefs

Cf. A003418, A014963, A056040.

Programs

Maple

a := proc(n) local A014963, k;
A014963 := proc(n) if n < 2 then 1 else numtheory[factorset](n);
if 1 < nops(%) then 1 else op(%) fi fi end;
mul(A014963(k)*(k/2)^((-1)^k), k=1..n)/2^n end;
# Also:
A180000 := proc(n) local lcm, sf;
lcm := ilcm(seq(i,i=1..n));
sf := n!/iquo(n,2)!^2;
lcm/sf end;

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 *)

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)

Formula

a(n) = 2^(-n)*Product_{1<=k<=n} A014963(k)*(k/2)^((-1)^k).