%I #30 Aug 02 2019 03:46:23
%S 1,1,1,1,2,2,24,24,384,3456,276480,276480,955514880,955514880,
%T 428070666240,866843099136000,1775294667030528000,1775294667030528000,
%U 331312591939905257472000,331312591939905257472000,339264094146462983651328000000
%N a(n) = Product_{k=1..n} Product_{i=1..k-1} gcd(k,i).
%C The order of the primes in the prime factorization of a(n) is given by
%C ord_{p}(a(n)) = (1/2)*Sum_{i>=1} floor(n/p^i)*(floor(n/p^i)-1).
%C 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
%C 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
%H Charles R Greathouse IV, <a href="/A224479/b224479.txt">Table of n, a(n) for n = 0..97</a>
%H OEIS Wiki, <a href="/wiki/Generalizations_of_the_factorial#Formulae_for_GCD_matrix_generalization_of_the_factorial">Generalizations of the factorial</a>
%F a(n) = Product_{k=1..n} Product_{d divides k, d < k} d^phi(k/d).
%F n! * a(n)^2 = A092287(n).
%F a(n)/a(n-1) = A051190(n) for n >= 1.
%F a(n) = sqrt(A092287(n) / n!). - _Daniel Forgues_, Apr 14 2013
%p A224479 := proc(n) local h, k, d;
%p mul(mul(d^phi(k/d), d = divisors(k) minus {k}), k = 1..n) end:
%p seq(A224479(i), i = 0..20);
%t 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 *)
%o (Sage) def A224479(n):
%o R = 1;
%o for p in primes(n):
%o s = 0; r = n
%o while r > 0 :
%o r = r//p
%o s += r*(r-1)
%o R *= p^(s/2)
%o return R
%o [A224479(i) for i in (0..20)]
%o (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
%Y Cf. A051190, A092287.
%K nonn
%O 0,5
%A _Peter Luschny_, Apr 07 2013