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The article derives the formula for the sum of the $k$-th powers of positive integers from $1$ to $n$ in the form of a polynomial in the variable $n$. The determination of the coefficients $a_{k,j}$ of the polynomial (a two-parametric problem) is converted into the determination of the members of a progression $B_p% , so called Bernoulli numbers (a one-parametric problem), and a recurrent formula for these numbers is derived. Then, mutual divisibility of the polynomials is examined for different values of $k$, and Nikomachos theorem is mentioned as a special case.
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