Problème noA2845 Diophante (février 2021) SoientS =Pn
k=2b√k
ncetS0 =Pn
j=2blogjnc.
n+S=n+ card{(k, j)∈N2 |2≤k≤n, 1≤j≤ √k n}
=n+ card{(k, j)∈N2 |2≤k≤n, 1≤j, jk≤n}
=n+ card{(k, j)∈N2 |2≤k≤n, 1≤j≤n, jk≤n}
= card{(k, j)∈N2 |1≤k≤n, 1≤j ≤n, jk≤n}
et
n+S0=n+ card{(k, j)∈N2 |2≤j≤n, 1≤k≤logjn}
=n+ card{(k, j)∈N2 |2≤j≤n, 1≤k, jk≤n}
=n+ card{(k, j)∈N2 |2≤j≤n, 1≤k≤n, jk≤n}
= card{(k, j)∈N2 |1≤j≤n, 1≤k≤n, jk ≤n}
donc S=S0.