Let us define the following recursive function involving the sum of divisors function $\sigma(n)$:
\begin{array}{ l } r(n,1)=\sigma(n) \\ r(n,2)=\sum_{d|n}r(d,1) \\ r(n,3)=\sum_{d|n}r(d,2) \\ \ldots \\ r(n,k)=\sum_{d|n}r(d,k-1) \\ \end{array}
As well explained here $r(n,k)$ is the Dirichlet convolution of $k$ copies of $\sigma$, which means it is multiplicative. Now let us define the sum $s(p,m,k)$ involving the Figurate Number $P_k(m)=\binom{m+k-1}{k}$ as follows:
$$ s(p,m,k)=\sum _{i=1}^{m+1} P_k(i)\cdot p^{m-i+1} $$
By setting for example $m=4$ and $k=2$ we obtain $s(p,4,2)=p^4+2 p^3+3 p^2+4 p+5$. By setting $m=4$ and $k=3$ we obtain $s(p,4,3)=p^4+3 p^3+6 p^2+10 p+15$.
My question:
How can we show that if $k\in{2,3}$ and $p$ is a prime and $p^m$ divide the integer $n$ exactly $p^m||n$, then $s(p,m,k)$ divides $r(n,k)$? Does there exist another $k$ satisfying this divisibility (I scanned $k\le600$ and did not find any)?
My investigations:
I coded this observation as follows and made cross checks:
rn = Compile[{{n, _Integer}, {k, _Integer}}, Total[Nest[Catenate@*Divisors, {n}, k]]];
s[p_, m_, k_] := Sum[p^(m - i + 1)*PolygonalNumber[k, i], {i, 1, m + 1}];
Print[FullSimplify[rn[7^5*2, 3]/s[7, 5, 3]]];
By choosing $n$ as a product of a prime power and another integer (which is coprime to this prime power), for example $n=7^5\cdot2=p^m\cdot2$, we may verify this behavior. This divisibility observation seems to hold for any case where $p$ is a prime and $k\in{2,3}$.
Plotting both functions, $r(n,k)$ and $s(p,m,k)$, with $m=5$, $k=3$ and $n=p^m\cdot2$ via primes = Table[Prime[p], {p, 80}]; DiscretePlot[{rn[p^m*2, k], s[p, m, k]}, {p, primes}, ExtentSize -> Full] generates the following chart:
The data for this case ($m=5$, $k=3$) looks as follows and we see that if $p$ is a prime then the divisibility holds:
\begin{array}{cccc} p & r(p^m,k) & s(p,m,k) & \frac{r(p^m,k)}{s(p,m,k)} \\ 1 & 35 & 1 & \frac{1}{35} \\ 2 & 99 & 99 & 1 \\ 3 & 261 & 261 & 1 \\ 4 & 599 & 1981 & \frac{1981}{599} \\ 5 & 1215 & 1215 & 1 \\ 6 & 2235 & 25839 & \frac{8613}{745} \\ 7 & 3809 & 3809 & 1 \\ 8 & 6111 & 32647 & \frac{32647}{6111} \\ 9 & 9339 & 22113 & \frac{7371}{3113} \\ 10 & 13715 & 120285 & \frac{24057}{2743} \\ 11 & 19485 & 19485 & 1 \\ 12 & 26919 & 517041 & \frac{57449}{2991} \\ 13 & 36311 & 36311 & 1 \\ 14 & 47979 & 377091 & \frac{41899}{5331} \\ 15 & 62265 & 317115 & \frac{21141}{4151} \\ 16 & 79535 & 524097 & \frac{524097}{79535} \\ 17 & 100179 & 100179 & 1 \\ 18 & 124611 & 2189187 & \frac{729729}{41537} \\ 19 & 153269 & 153269 & 1 \\ 20 & 186615 & 2406915 & \frac{53487}{4147} \\ 21 & 225135 & 994149 & \frac{110461}{25015} \\ 22 & 269339 & 1929015 & \frac{1929015}{269339} \\ 23 & 319761 & 319761 & 1 \\ 24 & 376959 & 8520867 & \frac{2840289}{125653} \\ 25 & 441515 & 762925 & \frac{152585}{88303} \\ 26 & 514035 & 3594789 & \frac{399421}{57115} \\ 27 & 595149 & 1793557 & \frac{1793557}{595149} \\ 28 & 685511 & 7545629 & \frac{7545629}{685511} \\ 29 & 785799 & 785799 & 1 \\ 30 & 896715 & 31394385 & \frac{697653}{19927} \\ 31 & 1018985 & 1018985 & 1 \\ \vdots & \vdots & \vdots & \vdots \\ 78 & 38476011 & 938239929 & \frac{10784367}{442253} \\ 79 & 40467449 & 40467449 & 1 \\ 80 & 42535215 & 636777855 & \frac{14150619}{945227} \\ \end{array}