p-adic valuation of Harmonic sums

\displaystyle \begin{array}{c|c} n & H_{n} \\ \hline 1 & 1 \\ 2 & 3 / 2 \\ 3 & 11 / 2 \cdot 3 \\ 4 & 25 / 2^{2} \cdot 3 \\ 5 & 137 / 2^{2} \cdot 15 \\ 6 & 49 / 2^{2} \cdot 5 \\ 7 & 363 / 2^{2} \cdot 35 \\ 8 & 761 / 2^{3} \cdot 35 \\ 9 & 7129 / 2^{3} \cdot 315 \\ 10 & 7381 / 2^{3} \cdot 2520 \\ 11 & 83711 / 2^{3} \cdot 3465\\ 12 & 86021/2^{3} \cdot 3465\\ 13 & 1145993/2^{3}\cdot 45045 \end{array}

Look at the p adic-valuation of Harmonic sums.
\displaystyle p=2 : For 2-valuation the denominator contains a power of 2 which equals the highest power of 2 less than \displaystyle n . When you take common denominators and compute the numerator- you see that all terms except this power of 2 will gives even contributions and this terms gives an odd contribution. Hence the numerator is odd.

\displaystyle \begin{array}{c|c} n & H_{n} \\ \hline 1 & 1 \\ 2 & 3 / 2 \\ 3 & 11 / 2 \cdot 3 \\ 4 & 5^{2} / 2^{2} \cdot 3 \\ 5 & 137 / 2^{2} \cdot 3 \cdot 5 \\ 6 & 7^{2} / 2^{2} \cdot 5 \\ 7 & 3 \cdot 11^{2} / 2^{2} \cdot 5 \cdot 7 \\ 8 & 761 / 2^{3} \cdot 5 \cdot 7 \\ 9 & 7129 / 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \\ 10 & 11^{2} \cdot 61 / 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \\ 11 & 97 \cdot 863 / 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\\ 12 & 132\cdot 509/2^{3} \cdot 3^2\cdot 5\cdot 7\cdot 11\\ 13 & 29 \cdot 43\cdot 919/2^{3}\cdot 3^2\cdot 5\cdot 7\cdot 11\cdot 13\end{array}

What do we know about evaluations for other primes?

Posted in $.

Leave a comment