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A334795
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a(n) = Product_{d|n} lcm(d, tau(d)) where tau(k) is the number of divisors of k (A000005).
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1
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1, 2, 6, 24, 10, 144, 14, 192, 54, 400, 22, 20736, 26, 784, 3600, 15360, 34, 23328, 38, 288000, 7056, 1936, 46, 3981312, 750, 2704, 5832, 790272, 58, 207360000, 62, 1474560, 17424, 4624, 19600, 120932352, 74, 5776, 24336, 92160000, 82, 796594176, 86, 3066624
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OFFSET
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1,2
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COMMENTS
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If p is an odd prime, a(p) = 2*p.
If p is a prime > 3, a(p^2) = 6*p^3.
If p and q are distinct odd primes, a(p*q) = 16*p^2*q^2. (End)
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LINKS
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FORMULA
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a(p) = 2p for p = odd primes (A065091).
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EXAMPLE
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a(6) = lcm(1, tau(1)) * lcm(2, tau(2)) * lcm(3, tau(3)) * lcm(6, tau(6)) = lcm(1, 1) * lcm(2, 2) * lcm(3, 2) * lcm(6, 4) = 1 * 2 * 6 * 12 = 144.
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MAPLE
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g:= d -> ilcm(d, numtheory:-tau(d)):
f:= n -> mul(g(d), d = numtheory:-divisors(n)):
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MATHEMATICA
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a[n_] := Product[LCM[d, DivisorSigma[0, d]], {d, Divisors[n]}]; Array[a, 100] (* Amiram Eldar, May 12 2020 *)
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PROG
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(Magma) [&*[LCM(d, #Divisors(d)): d in Divisors(n)]: n in [1..100]]
(PARI) a(n) = my(d=divisors(n)); prod(k=1, #d, lcm(d[k], numdiv(d[k]))); \\ Michel Marcus, May 12 2020
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CROSSREFS
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Cf. A334782 (Sum_{d|n} lcm(d, tau(d))), A334664 (Product_{d|n} gcd(d, tau(d))).
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KEYWORD
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AUTHOR
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STATUS
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approved
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