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

1,2

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

Links

Table of n, a(n) for n=1..19.

Example

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.

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.

Sequence in context: A224352 A224194 A224011 * A009980 A041613 A255821

Adjacent sequences: A300354 A300355 A300356 * A300358 A300359 A300360

Keyword

nonn

Author

Jon E. Schoenfield, Mar 03 2018

Status

approved