A300357 a(n) is the smallest number whose number of divisors is the n-th odd square.

1, 36, 1296, 46656, 44100, 60466176, 2176782336, 1587600, 2821109907456, 101559956668416, 57153600, 131621703842267136, 1944810000, 341510400, 6140942214464815497216, 221073919720733357899776, 74071065600, 70013160000, 10314424798490535546171949056

Offset: 1

Author

Jon E. Schoenfield, Mar 03 2018

Keywords

• nonn

Comments

Equivalently, a(n) is the smallest number having exactly (2n-1)^2 divisors. (Since the number of divisors is odd, each term is necessarily a square.)

Subsequence of A025487.

Bisection of A061707. - Michel Marcus, Mar 04 2018

Examples

			For n=2, the n-th odd square is (2n-1)^2 = (2*2-1)^2 = 9. Each number having exactly 9 divisors is of one of the forms p^8 or p^2*q^2 where p and q are distinct primes. The smallest number of the form p^8 is 2^8=256, but the smallest of the form p^2*q^2 is 2^2*3^2 = 36, so a(2)=36.
For n=5, the n-th odd square is 81. Each number having exactly 81 divisors is of one of the forms p^80, p^26*q^2, p^8*q^8, p^8*q^2*r^2, or p^2*q^2*r^2*s^2, where p, q, r, and s are distinct primes. Since the exponents in each form as written above are in nonincreasing order, the smallest number of each form is obtained by assigning the first few primes in increasing order to p, q, r, and s, i.e., p=2, q=3, r=5, and s=7. The smallest resulting number is 2^2*3^2*5^2*7^2 = 44100, so a(5)=44100.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..644

Crossrefs

Cf. A000005 (number of divisors of n), A000290 (squares), A016754 (odd squares), A005179 (smallest number with exactly n divisors), A025487 (products of primorials), A061707.

Formula

a(n) = A005179(A016754(n-1)) = A005179((2*n-1)^2). - Amiram Eldar, Jan 23 2025