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Omni-MATH / Call a sequence of positive integers {a_(n)} good if for any distinct positive integers m,n, one has gcd(m,n)∣a_(m)²+a_(n)² and gcd(a_(m),a_(n))∣m²+n². Call a positive integer a to…

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problem

Call a sequence of positive integers

\{a_n\}

good if for any distinct positive integers

m,n

, one has

\gcd(m,n) \mid a_m^2 + a_n^2 \text{ and } \gcd(a_m,a_n) \mid m^2 + n^2.

Call a positive integer

a

to be

k

-good if there exists a good sequence such that

a_k = a

. Does there exists a

k

such that there are exactly

2019

k

-good positive integers?

Plain-text mathematical notation (without MathML)

Call a sequence of positive integers {a_(n)} good if for any distinct positive integers m,n, one has 
gcd(m,n)∣a_(m)²+a_(n)² and gcd(a_(m),a_(n))∣m²+n².
Call a positive integer a to be k-good if there exists a good sequence such that a_(k)=a. Does there exists a k such that there are exactly 2019 k-good positive integers?

Original LaTeX notation

Call a sequence of positive integers $\{a_n\}$ good if for any distinct positive integers $m,n$, one has 
$$\gcd(m,n) \mid a_m^2 + a_n^2 \text{ and } \gcd(a_m,a_n) \mid m^2 + n^2.$$
Call a positive integer $a$ to be $k$-good if there exists a good sequence such that $a_k = a$. Does there exists a $k$ such that there are exactly $2019$ $k$-good positive integers?

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task_id: 670f5f3c-1a34-5183-9039-a463e472f60b
task_key: test--670f5f3c-1a34-5183-9039-a463e472f60b
task_revision_id: 3
config / split: / test

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