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A224479
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a(n) = Product_{k=1..n} Product_{i=1..k-1} gcd(k,i).
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4
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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
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0,5
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COMMENTS
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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
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LINKS
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FORMULA
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a(n) = Product_{k=1..n} Product_{d divides k, d < k} d^phi(k/d).
a(n)/a(n-1) = A051190(n) for n >= 1.
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MAPLE
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mul(mul(d^phi(k/d), d = divisors(k) minus {k}), k = 1..n) end:
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MATHEMATICA
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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 *)
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PROG
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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
(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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