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 A300357 a(n) is the smallest number whose number of divisors is the n-th odd square. 2

%I

%S 1,36,1296,46656,44100,60466176,2176782336,1587600,2821109907456,

%T 101559956668416,57153600,131621703842267136,1944810000,341510400,

%U 6140942214464815497216,221073919720733357899776,74071065600,70013160000,10314424798490535546171949056

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

%C 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.)

%C Subsequence of A025487.

%C Bisection of A061707. - _Michel Marcus_, Mar 04 2018

%e 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.

%e 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.

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

%K nonn

%O 1,2

%A _Jon E. Schoenfield_, Mar 03 2018

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Last modified November 15 11:18 EST 2019. Contains 329144 sequences. (Running on oeis4.)