Let $\sigma(n)=\sum_{d\mid n}d$ the sum of divisors function, and $s(n)=\sum_{k=1}^n n \operatorname{mod} k$ the sum of remainders function, then for $n\geq 2$ $$\sigma(n)+s(n)=s(n-1)+2n-1$$ holds, see for example the proof of Theorem 1, in Spivey, The Humble Sum of Remainders Function, Mathematics Magazine Vol. 78, No. 4, (2005). I am interested in deduce new identities involving these functions, I believe that the sum of remainders function is an interesting arithmetic function. This night I tried this, but I believe that my deductions isn't so interesting as deduced by Spivey, because mine are artificious.
My attempt was inspiring in such calculations to combine an arithmetical identity, in my case the sum of a geometric series with previous identity. In this Question, I am asking if my deductions were rights and if you can see a different approach to get more identities:
For $k\geq 1$ then a specialisation provide us $$\sigma(n^k)+s(n^k)=s(n^k-1)+2n^k-1,$$ and since $n>1$ the sum of the geometric series $$\sum_{k=1}^m n^k=\frac{n^{m+1}-n}{n-1}$$ then $$\sum_{k=1}^m\sigma(n^k)+\sum_{k=1}^m s(n^k)-\sum_{k=1}^m s(n^k-1)=2\frac{n^{m+1}-n}{n-1}-m.$$ And a new specialisation of this last identity, taking now $m=d$ as the divisors of $n=N$ (this last is only notation) and taking the sum for $d\mid N$ provide us the first identity $$\sum_{d\mid N}\sum_{k=1}^d\left(\sigma(N^k)+s(N^k)-s(N^k-1)\right)=2\frac{N}{N-1}\left(\sum_{d\mid N} N^d-\sigma_0(N)\right)-\sigma(N).$$ Here with $\sigma_0(N)=\sum_{d\mid N}1$ we are denoting the divisor-counting function. But as I was said I don't know if previous identity in LHS is interesting because is artificious (a tautology with respect RHS).
As a second approach, from our first deduction, when I multiplied by $-2$ and after I've added $-1$ then I've deduced this second identity $$-1-2\sum_{k=1}^m\left(\sigma(n^k)+s(n^k)-s(n^k-1)\right)=-4\frac{n^{m+1}-n}{n-1}+\sigma(m)+s(m)-s(m-1).$$
Question. a) Are there mistakes in the deduced identities? b) Can you deduce a new identity using this idea?, I say with specialisation for different sequences of integers and combining with an arithmetic identity or asymptotic identities, that provide us some more interesting identity involving the sum of remainders function and the sum of divisors function.
My purpose is edit good posts in this Mathematics Stack Exchange, thus those answers for b) will be the best reference for all us in this site. Many thanks.