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

1,1

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.

Links

Table of n, a(n) for n=1..45.

Example

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).

Prog

(PARI) r()={my(i=if(!#s2, 0, #s2[#s2])%#s1[1]+1); if(i==1, listput(s2, List())); for(i2=0, s1[#s1][i], listput(s2[#s2], i2); if(i<#s1[#s1], r(), p=prod(i3=1, #s2[#s2], v[1][i3]^s2[#s2][i3]); if(p>1 && (!#f || p>=f[#f]), listput(f, p); listput(s1, s1[#s1]); for(i3=1, #s2[#s2], s1[#s1][i3]-=s2[#s2][i3]); if(vecsum(s1[#s1])>0, r(), if(issquare(sum(i3=1, #f, f[i3]^2)), d=sqrtint(sum(i3=1, #f, f[i3]^2)); mapput(m, d, if(mapisdefined(m, d), mapget(m, d), 0)+1))); listpop(s1); listpop(f))); listpop(s2[#s2])); if(i==1, listpop(s2))}
for(t=2, 1000, v=Vec(factor(t)); s1=List([v[2]]); s2=List(); f=List(); m=Map(); r(); m2=Vec(Mat(m)); for(f2=1, #m2[1]-1, if(t%m2[1][f2]==0, print(t); break)))

Crossrefs

Subset of A380436.

Sequence in context: A097762 A297610 A083469 * A224160 A224411 A260357

Adjacent sequences: A380434 A380435 A380436 * A380439 A380440 A380441

Keyword

nonn,new

Author

Charles L. Hohn, Jan 24 2025

Status

approved