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.

16, 108, 192, 240, 256, 300, 576, 768, 864, 960, 1024, 1080, 1152, 1200, 1260, 1296, 1344, 1350, 1458, 1500, 1680, 1836, 2016, 2160, 2304, 2400, 2592, 2688, 2700, 2772, 2800, 2880, 2916, 3024, 3240, 3264, 3344, 3510, 3600, 3780, 3840, 4096, 4400, 4608, 4800

Offset: 1

Author

Charles L. Hohn, Jan 24 2025

Keywords

• nonn

Comments

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.

Examples

			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).
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).
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).
		

Crossrefs

Subset of A380436.

Programs

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 && dCharles L. Hohn, Mar 09 2025