cp's OEIS Frontend

A380437 Integers with at least 1 proper factorization for which the sum of the squares of the factors is a square, whose square root is also a factor of that number.

%I A380437 #8 Mar 16 2025 18:24:48
%S A380437 16,108,192,240,256,300,576,768,864,960,1024,1080,1152,1200,1260,1296,
%T A380437 1344,1350,1458,1500,1680,1836,2016,2160,2304,2400,2592,2688,2700,
%U A380437 2772,2800,2880,2916,3024,3240,3264,3344,3510,3600,3780,3840,4096,4400,4608,4800
%N A380437 Integers with at least 1 proper factorization for which the sum of the squares of the factors is a square, whose square root is also a factor of that number.
%C A380437 This sequence includes all values b^(c^2) for each integer b >= 2 and c >= 2 where the prime factors of c are equal to or a subset of the prime factors of b. In these cases, c*b, which is the square root of the sum of c^2 squares of b, will always be a factor of b^(c^2). E.g. b=2 and c=2 (both with only {2} as their prime factor) gives 2^(2^2) = 16 (a(1)), which can be factored as {2, 2, 2, 2}, for which sqrt(2^2 + 2^2 + 2^2 + 2^2)=4, also a factor of 16.
%e A380437 a(1) = 16: {2, 2, 2, 2} (2 * 2 * 2 * 2 = 16 and sqrt(2^2 + 2^2 + 2^2 + 2^2) = 4, which is also a factor of 16).
%e A380437 a(2) = 108: {3, 6, 6} (3 * 6 * 6 = 108 and sqrt(3^2 + 6^2 + 6^2) = 9, which is also a factor of 108).
%e A380437 a(3) = 192: {2, 2, 2, 4, 6} (2 * 2 * 2 * 4 * 6 = 192 and sqrt(2^2 + 2^2 + 2^2 + 4^2 + 6^2) = 8, which is also a factor of 192).
%o A380437 (PARI) a380437_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(r==1, my(t=sum(i=1, #f, f[i]^2)); return(if(issquare(t) && x%sqrtint(t)==0, 1, 0))); my(d, c=0); fordiv(r, d, if(d==1 || d==x || (#f && d<f[#f]), next); listput(f, d); c+=a380437_count(x, f); listpop(f)); return(c)} \\ _Charles L. Hohn_, Mar 09 2025
%Y A380437 Subset of A380436.
%K A380437 nonn
%O A380437 1,1
%A A380437 _Charles L. Hohn_, Jan 24 2025