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A130279 Smallest number having exactly n square divisors. 15
1, 4, 16, 36, 256, 144, 4096, 576, 1296, 2304, 1048576, 3600, 16777216, 36864, 20736, 14400, 4294967296, 32400, 68719476736, 57600, 331776, 9437184, 17592186044416, 129600, 1679616, 150994944, 810000, 921600, 72057594037927936 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
A046951(a(n)) = n and A046951(m) <> n for m < a(n);
all terms are smooth squares: if prime(k) is a factor of a(n) then also prime(i) are factors, i<k;
a(p) = 2^(2*(p-1)) for primes p;
if prime(j) is the greatest prime factor of a(n) then a(2*n) = a(n)*prime(j+1)^2;
A001221(a(n)) = A122375(n); A001222(a(n)) = 2*A122376(n).
a(n+1) is the smallest nonsquarefree number m such that Diophantine equation S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = m has exactly n solutions, for n >= 0 (A353282); example: a(4) = 36 and 36 is the smallest number m such that equation S(x,y) = m has exactly 3 solutions: (9,1), (8,2), (5,5). - Bernard Schott, Apr 13 2022
a(n) is the square of the smallest integer having exactly n divisors (see formula with proof). - Bernard Schott, Oct 01 2022
LINKS
Eric Weisstein's World of Mathematics, Smooth Number
FORMULA
From Bernard Schott, Oct 01 2022: (Start)
a(n) = A005179(n)^2.
Proof: Suppose a(n) = Product p_i^(2*e_i), where the p_i are primes. Then the n square divisors are all of the form d = Product p_i^(2*k_i) with 0 <= k_i <= e_i. As a(n) = Product (p_i^e_i)^2 = (Product (p_i^e_i))^2, we get that sqrt(a(n)) = Product (p_i^e_i). This is the prime decomposition of sqrt(a(n)). As there is a bijection between prime factors p_i^(2*k_i) and (p_i^k_i), there is also bijection between square divisors of a(n) and divisors of sqrt(a(n)). We conclude that sqrt(a(n)) is the smallest integer that has exactly n divisors. (End)
PROG
(PARI) a(n) = my(k=1); while(sumdiv(k, d, issquare(d)) != n, k++); k; \\ Michel Marcus, Jul 15 2019
CROSSREFS
Cf. A357450 (similar, but with odd squares divisors).
Sequence in context: A223403 A263385 A368215 * A030158 A054246 A173545
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, May 20 2007
STATUS
approved

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Last modified March 29 01:36 EDT 2024. Contains 371264 sequences. (Running on oeis4.)