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%I #61 Aug 28 2024 16:36:50
%S 1,6,12,168,30,9,56,960,351,450,132,6048,182,294,1800,158720,306,
%T 25272,380,84000,14112,1089,552,414720,11625,7098,29160,32928,870,
%U 101250,992,2064384,17424,15606,58800,917070336,1406,5415,85176,11520000,1722,777924,1892
%N a(n) = lcm(tau(n), sigma(n), pod(n)) / gcd(tau(n), sigma(n), pod(n)) where tau(k) is the number of divisors of k (A000005), sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).
%C a(n) is also lcm(n, tau(n), sigma(n), pod(n)) / gcd(tau(n), sigma(n), pod(n)).
%F a(n) = lcm(n, tau(n), sigma(n), pod(n)) / gcd(tau(n), sigma(n), pod(n)).
%F a(n) = A336723(n) / A336722(n).
%F a(p) = p * (p+1) for p = primes.
%e a(6) = lcm(tau(6), sigma(6), pod(6)) / gcd(tau(6), sigma(6), pod(6)) = lcm(4, 12, 36) / gcd(4, 12, 36) = 36 / 4 = 9.
%t a[n_] := LCM @@ (t = {(d = DivisorSigma[0, n]), n^(d/2), DivisorSigma[1, n]}) / GCD @@ t; Array[a, 50] (* _Amiram Eldar_, Aug 31 2020 *)
%o (Magma) [LCM([#Divisors(n), &+Divisors(n), &*Divisors(n)]) / GCD([#Divisors(n), &+Divisors(n), &*Divisors(n)]): n in [1..100]];
%o (PARI) a(n) = my(f=factor(n), v=[numdiv(f), sigma(f), vecprod(divisors(f))]); lcm(v)/gcd(v); \\ _Michel Marcus_, Aug 31 2020
%Y Cf. A336722, A336723, A337323.
%Y Cf. A334985 (lcm(n, tau(n), sigma(n), pod(n)) / gcd(n, tau(n), sigma(n), pod(n))).
%Y Cf. A000005, A000203, A007955.
%K nonn
%O 1,2
%A _Jaroslav Krizek_, Aug 31 2020