A220027 a(n) = product(i >= 0, P(n, i)^(2^i)) where P(n, i) = product(p prime, n/2^(i+1) < p <= n/2^i).

1, 1, 2, 6, 12, 60, 180, 1260, 5040, 5040, 25200, 277200, 2494800, 32432400, 227026800, 227026800, 3632428800, 61751289600, 61751289600, 1173274502400, 29331862560000, 29331862560000, 322650488160000, 7420961227680000, 601097859442080000, 601097859442080000

Offset: 0

Author

Peter Luschny, Mar 30 2013

Keywords

• nonn

Comments

a(n) are the partial products of A219964(n).

a(n) divides n!, n!/a(n) = 1, 1, 1, 1, 2, 2, 4, 4, 8, 72, 144, 144, 192...

The swinging factorial (A056040) divides a(n), a(n)/n$ = 1, 1, 1, 1, 2,...

The primorial of n (A034386) divides a(n), a(n)/n# = 1, 1, 1, 1, 2, 2, 6,..

If p^k is the largest power of a prime dividing a(n) then k is 2^n for some n >= 0.

a(n) / A055773(n) is the largest square dividing a(n), a(n) / A055773(n) = A008833(a(n)).

Crossrefs

Cf. A055773.

Programs

Maple

a := proc(n) local k; `if`(n < 2, 1,
mul(k, k = select(isprime, [$iquo(n, 2)+1..n]))*a(iquo(n,2))^2) end:
seq(a(i), i=0..25);

Sage

def a(n) :
    if n < 2 : return 1
    return mul(k for k in prime_range(n//2+1,n+1))*a(n//2)^2
[a(n) for n in (0..25)]