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

%I A300357 #16 Jan 23 2025 03:43:06
%S A300357 1,36,1296,46656,44100,60466176,2176782336,1587600,2821109907456,
%T A300357 101559956668416,57153600,131621703842267136,1944810000,341510400,
%U A300357 6140942214464815497216,221073919720733357899776,74071065600,70013160000,10314424798490535546171949056
%N A300357 a(n) is the smallest number whose number of divisors is the n-th odd square.
%C A300357 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 A300357 Subsequence of A025487.
%C A300357 Bisection of A061707. - _Michel Marcus_, Mar 04 2018
%H A300357 Amiram Eldar, <a href="/A300357/b300357.txt">Table of n, a(n) for n = 1..644</a>
%F A300357 a(n) = A005179(A016754(n-1)) = A005179((2*n-1)^2). - _Amiram Eldar_, Jan 23 2025
%e A300357 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 A300357 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 A300357 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 A300357 nonn
%O A300357 1,2
%A A300357 _Jon E. Schoenfield_, Mar 03 2018