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A130279 Smallest number having exactly n square divisors.

%I A130279 #31 Feb 16 2025 08:33:06
%S A130279 1,4,16,36,256,144,4096,576,1296,2304,1048576,3600,16777216,36864,
%T A130279 20736,14400,4294967296,32400,68719476736,57600,331776,9437184,
%U A130279 17592186044416,129600,1679616,150994944,810000,921600,72057594037927936
%N A130279 Smallest number having exactly n square divisors.
%C A130279 A046951(a(n)) = n and A046951(m) <> n for m < a(n);
%C A130279 all terms are smooth squares: if prime(k) is a factor of a(n) then also prime(i) are factors, i<k;
%C A130279 a(p) = 2^(2*(p-1)) for primes p;
%C A130279 if prime(j) is the greatest prime factor of a(n) then a(2*n) = a(n)*prime(j+1)^2;
%C A130279 A001221(a(n)) = A122375(n); A001222(a(n)) = 2*A122376(n).
%C A130279 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
%C A130279 a(n) is the square of the smallest integer having exactly n divisors (see formula with proof). - _Bernard Schott_, Oct 01 2022
%H A130279 Amiram Eldar, <a href="/A130279/b130279.txt">Table of n, a(n) for n = 1..100</a>
%H A130279 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SmoothNumber.html">Smooth Number</a>
%F A130279 From _Bernard Schott_, Oct 01 2022: (Start)
%F A130279 a(n) = A005179(n)^2.
%F A130279 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)
%o A130279 (PARI) a(n) = my(k=1); while(sumdiv(k, d, issquare(d)) != n, k++); k; \\ _Michel Marcus_, Jul 15 2019
%Y A130279 Cf. A001221, A005179, A046951, A046952, A122375, A122376, A353282.
%Y A130279 Cf. A357450 (similar, but with odd squares divisors).
%K A130279 nonn
%O A130279 1,2
%A A130279 _Reinhard Zumkeller_, May 20 2007