A224479 - OEIS

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

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OFFSET

0,5

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

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

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 *)

PROG

(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)]

(PARI) a(n)=prod(k=1, n, my(s=1); fordiv(k, d, d<k && s*=d^eulerphi(k/d)); s) \\ Charles R Greathouse IV, Jun 27 2013