A336723 - OEIS

%I #19 Dec 21 2024 15:20:18

%S 1,6,12,168,30,36,56,960,351,900,132,12096,182,1176,1800,158720,306,

%T 75816,380,168000,14112,4356,552,1658880,11625,14196,29160,65856,870,

%U 810000,992,2064384,17424,31212,58800,917070336,1406,21660,85176,23040000,1722,6223392

%N a(n) = lcm(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) = pod(n) for numbers n: 1, 6, 30, 66, 84, 102, 120, 210, 270, 318, 330, 420, 462, 510, 546, 570, 642, ...

%F a(p) = p^2 + p for p = primes (A000040).

%e a(6) = lcm(tau(6), sigma(6), pod(6)) = lcm(4, 12, 36) = 36.

%t a[n_] := LCM @@ {(d = DivisorSigma[0,n]), DivisorSigma[1, n], n^(d/2)}; Array[a, 50] (* _Amiram Eldar_, Aug 01 2020 *)

%o (Magma) [LCM([#Divisors(n), &+Divisors(n), &*Divisors(n)]): n in [1..100]];

%o (PARI) a(n) = my(d=divisors(n)); lcm([#d, vecsum(d), vecprod(d)]); \\ _Michel Marcus_, Aug 12 2020

%Y Cf. A009278 (lcm(tau(n), sigma(n))), A324528 (lcm(tau(n), pod(n))), A324529 (lcm(sigma(n), pod(n))).

%Y Cf. A000005 (tau(n)), A000203 (sigma(n)), A007955 (pod(n)), A336722 (gcd(tau(n), sigma(n), pod(n))).

%Y Cf. A277521 (numbers k such that a(k) = pod(k) and simultaneously A336722(k) = tau(k)).

%K nonn,changed

%O 1,2

%A _Jaroslav Krizek_, Aug 01 2020