A224479 a(n) = Product_{k=1..n} Product_{i=1..k-1} gcd(k,i).

1, 1, 1, 1, 2, 2, 24, 24, 384, 3456, 276480, 276480, 955514880, 955514880, 428070666240, 866843099136000, 1775294667030528000, 1775294667030528000, 331312591939905257472000, 331312591939905257472000, 339264094146462983651328000000

Offset: 0

Author

Peter Luschny, Apr 07 2013

Keywords

• nonn

Comments

The order of the primes in the prime factorization of a(n) is given by

ord_{p}(a(n)) = (1/2)*Sum_{i>=1} floor(n/p^i)*(floor(n/p^i)-1).

Product of all entries of lower-left (excluding main diagonal) triangular submatrix of GCDs. Also the product of all entries of upper-right (excluding main diagonal) triangular submatrix of GCDs, since the matrix is symmetric. - Daniel Forgues, Apr 14 2013

a(n)^2 * n! gives A092287(n), where n! is the product of the main diagonal entries of the matrix. - Daniel Forgues, Apr 14 2013

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 0..97
• OEIS Wiki, Generalizations of the factorial

Crossrefs

Cf. A051190, A092287.

Programs

Maple

A224479 := proc(n) local h, k, d;
mul(mul(d^phi(k/d), d = divisors(k) minus {k}), k = 1..n) end:
seq(A224479(i), i = 0..20);

Mathematica

a[n_] := Product[ d^EulerPhi[k/d], {k, 1, n}, {d, Divisors[k] // Most}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 27 2013, after Maple *)

PARI

a(n)=prod(k=1,n,my(s=1);fordiv(k,d,dCharles R Greathouse IV, Jun 27 2013

Sage

def A224479(n):
    R = 1;
    for p in primes(n):
        s = 0; r = n
        while r > 0 :
            r = r//p
            s += r*(r-1)
        R *= p^(s/2)
    return R
[A224479(i) for i in (0..20)]

Formula

a(n) = Product_{k=1..n} Product_{d divides k, d < k} d^phi(k/d).

n! * a(n)^2 = A092287(n).

a(n)/a(n-1) = A051190(n) for n >= 1.

a(n) = sqrt(A092287(n) / n!). - Daniel Forgues, Apr 14 2013