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

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