A061236 Smallest number with prime(n)^3 divisors where prime(n) is n-th prime.

24, 900, 810000, 729000000, 590490000000000, 531441000000000000, 430467210000000000000000, 387420489000000000000000000, 313810596090000000000000000000000, 228767924549610000000000000000000000000000, 205891132094649000000000000000000000000000000

Offset: 1

Author

Labos Elemer, Jun 01 2001

Keywords

• nonn

Examples

			If p = 2, then d(128) = d(24) = d(30) = 8 and a(1) = 24 < 30 is the smallest.
If p = 5, then 2^124 > (2^24)*(3^4) > 30^4 = 810000 = a(3).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..123

Crossrefs

Cf. A000005, A005179, A003680, A030078, A037992, A061283, A061286, A061148, A061149, A061234.

Formula

For p = 2, 24 is the solution. If a prime p > 2, the suitable powers of 30 are the least solutions: a(n) = Min{x | d(x) = A000005(x) = p(n)^3} = 30^(prime(n)-1). d(2^(ppp-1)) = d(2^(pp-1)*3^(p-1)) = d(30^(p-1)) = p^3 and 2^(ppp-1) > 2^(pp-1)*3^(p-1) > 30^(p-1) holds if p > 2.

a(n) = A005179(A030078(n)) = A005179(prime(n)^3). - Amiram Eldar, Jan 23 2025

Extensions

a(10)-a(11) from Amiram Eldar, Jan 23 2025