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