A324987 - OEIS

%I #16 Sep 08 2022 08:46:24

%S 1,6,8,126,12,2304,16,7560,312,5184,24,8128512,28,9216,9216,1171800,

%T 36,21026304,40,27433728,16384,20736,48,234101145600,1116,28224,49920,

%U 65028096,60,110075314176,64,442940400,36864,46656,36864,60754075619328,76,57600,50176

%N a(n) = Product_{d|n} (tau(d)*sigma(d)) where tau(k) = the number of divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

%C n divides a(n) for n: 1, 2, 6, 8, 12, 18, 24, 28, 36, 40, 48, 54, 56, 72, 80, 84, 96, 108, 112, 117, ...

%F a(n) = Product_{d|n} tau(d) * Product_{d|n} sigma(d) = A211776(n) * A206032(n).

%F a(p) = 2*(p + 1) for p = primes (A000040).

%F a(n) = Product_{d|n} A064840(d). - _Antti Karttunen_, Mar 28 2019

%e a(6) = (tau(1)*sigma(1)) * (tau(2)*sigma(2)) * (tau(3)*sigma(3)) * (tau(6)*sigma(6)) = (1*1) * (2*3) * (2*4) * (4*12) = 2304.

%t Table[Product[DivisorSigma[0, k]*DivisorSigma[1, k], {k, Divisors[n]}], {n, 1, 50}] (* _Vaclav Kotesovec_, Mar 23 2019 *)

%o (Magma) [&*[NumberOfDivisors(d) * SumOfDivisors(d): d in Divisors(n)]: n in [1..100]]

%o (PARI) a(n) = my(d=divisors(n)); prod(k=1, #d, numdiv(d[k])*sigma(d[k])); \\ _Michel Marcus_, Mar 23 2019

%Y Cf. A000005, A000203, A064840, A206032, A211776, A324986.

%K nonn

%O 1,2

%A _Jaroslav Krizek_, Mar 23 2019