A374793 - OEIS

%I #17 Jul 20 2024 18:43:13

%S 2,1260,27935107200,29564884570506808579056000

%N a(n) is the largest k such that tau(k)^n >= k.

%C Let prime(j)# denote the product of the first j primes, A002110(j); then

%C   a(1)  = prime(1)# = 2,

%C   a(2)  = 6*prime(4)# = 1260,

%C   a(3)  = 2880*prime(8)# = 2.7935...*10^10,

%C   a(4)  = 907200*prime(16)# = 2.9564...*10^25,

%C   a(5) >= 259459200*prime(30)# = 8.2015...*10^54,

%C   a(6) >= 3238237626624000*prime(52)# = 3.4403...*10^111,

%C   a(7) >= 248818180782850398720000*prime(91)# = 5.4351...*10^218.

%e 27935107200 = 2^7 * 3^3 * 5^2 * 7^1 * 11^1 * 13^1 * 17^1 * 19^1,

%e so tau(27935107200) = (7+1)*(3+1)*(2+1)*(1+1)*(1+1)*(1+1)*(1+1)*(1+1) = 8*4*3*2*2*2*2*2 = 3072; 3072^3 = 28991029248 > 27935107200, and there is no larger number k such that tau(k)^3 >= k, so a(3) = 27935107200.

%Y Cf. A000005, A035033, A056757, A056758.

%K nonn,more

%O 1,1

%A _Jon E. Schoenfield_, Jul 20 2024