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Let $ x\geq 1 $ be a large number, let $ [x]=x-\{x\} $ be the largest integerfunction, and let $ \sigma(n)$ be the sum of divisors function. This notepresents the first proof of the asymptotic formula for the average order $\sum_{p\leq x}\sigma([x/p])=c_0x\log \log x+O(x) $ over the primes, where$c_0>0$ is a constant. More generally, $ \sum_{p\leqx}\sigma([x/(p+a)])=c_0x\log \log x+O(x) $ for any fixed integer $a$.

Sum of Divisors Function And The Largest Integer Function Over The Shifted Primes