why is

Sum of gcd(k + i * i , k + (i + 1) * (i + 1)) if i from 1 to 2 * k

are equal

to

( (Sum of gcd(i , 4 * k + 1) if i from 1 to 4 * k + 1 ) - 2 + 4 * k + 1 ) divides 2

Sum of gcd(k + i * i , k + (i + 1) * (i + 1)) if i from 1 to 2 * k

are equal

to

( (Sum of gcd(i , 4 * k + 1) if i from 1 to 4 * k + 1 ) - 2 + 4 * k + 1 ) divides 2