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It seems rather surprising that, given the Diophantine equation,

$$a^3+b^3+c^3 = n^3$$

then a good $\color{red}{99.8\%}$ of $n<1000000$ are solvable in positive integers $a,b,c$. (See the discussion in this MSE post.)

Oleg567 gave a list of the $1867$ unsolvable $n$ in that range. More than half were primes, while the divisibility by prime $p$ of the unsolvable composites are given by the table below. Oleg noticed a curious behavior:

\begin{array}{|l|c|c|c|c|} \hline p & \mbox{# divisible by } p & ...\; p^2 & ...\;p^3 \\ \hline 2 & \color{blue}{177} & 41 & 22\\ 3 & 31 &-&-\\ 5 & 53 &-&-\\ 7 & \color{blue}{218} & 52 & 11 \\ 11 & 43 &-&-\\ 13 & \color{blue}{300} & 46 & 4 \\ 17 & 47 &-&-\\ 23 & 31 &-&-\\ 31 & 34 &-&-\\ 37 & \color{blue}{82} &-&-\\ 43 & 39 &-&-\\ 47 & 29 &-&-\\ 59 & 24 &1&-\\ 61 & 13 &-&-\\ 73 & 16 &-&-\\ 79 & 48 &4&-\\ \hline \end{array}

Thus, in that range, there are just $31$ unsolvables of form,

$$a^3+b^3+c^3 = (3m)^3$$

but $218$ unsolvables,

$$a^3+b^3+c^3 = (7m)^3$$

and $300$,

$$a^3+b^3+c^3 = (13m)^3$$

Q: What could explain this "anomalous" behavior of $p=2^k,7^k,13^k$ for $k=1,2,3$ with respect to cubes?

It seems rather surprising that, given the Diophantine equation,

$$a^3+b^3+c^3 = n^3\tag1$$

then a good $\color{red}{99.8\%}$ of $n<1000000$ are solvable in positive integers $a,b,c$. (See the discussion in this MSE post.)

Oleg567 gave a list of the $1867$ unsolvable $n$ in that range. More than half were primes, while the divisibility by prime $p$ of the unsolvable composites are given by the table below. Oleg noticed a curious behavior:

\begin{array}{|l|c|c|c|c|} \hline p & \mbox{# divisible by } p & ...\; p^2 & ...\;p^3 \\ \hline 2 & \color{blue}{177} & 41 & 22\\ 3 & 31 &-&-\\ 5 & 53 &-&-\\ 7 & \color{blue}{218} & 52 & 11 \\ 11 & 43 &-&-\\ 13 & \color{blue}{300} & 46 & 4 \\ 17 & 47 &-&-\\ 23 & 31 &-&-\\ 31 & 34 &-&-\\ 37 & \color{blue}{82} &-&-\\ 43 & 39 &-&-\\ 47 & 29 &-&-\\ 59 & 24 &1&-\\ 61 & 13 &-&-\\ 73 & 16 &-&-\\ 79 & 48 &4&-\\ \hline \end{array}

Thus, in that range, there are just $31$ unsolvables of form,

$$a^3+b^3+c^3 = (3m)^3$$

but $218$ unsolvables,

$$a^3+b^3+c^3 = (7m)^3$$

and $300$,

$$a^3+b^3+c^3 = (13m)^3$$

Q: What could explain this "anomalous" behavior of $p=2^k,7^k,13^k$ for $k=1,2,3$ with respect to the cubic Diophantine equation $(1)$?

It seems rather surprising that, given the Diophantine equation,

$$a^3+b^3+c^3 = n^3$$

then a good $\color{red}{99.8\%}$ of $n<1000000$ are solvable in positive integers $a,b,c$. (See the discussion in this MSE post.)

Oleg567 gave a list of the $1867$ unsolvable $n$ in that range. More than half were primes, while the divisibility by prime $p$ of the unsolvable composites are given by the table below. Oleg noticed a curious behavior:

\begin{array}{|l|c|c|c|c|} \hline p & \mbox{# divisible by } p & ...\; p^2 & ...\;p^3 \\ \hline 2 & \color{blue}{177} & 41 & 22\\ 3 & 31 &-&-\\ 5 & 53 &-&-\\ 7 & \color{blue}{218} & 52 & 11 \\ 11 & 43 &-&-\\ 13 & \color{blue}{300} & 46 & 4 \\ 17 & 47 &-&-\\ 23 & 31 &-&-\\ 31 & 34 &-&-\\ 37 & \color{blue}{82} &-&-\\ 43 & 39 &-&-\\ 47 & 29 &-&-\\ 59 & 24 &1&-\\ 61 & 13 &-&-\\ 73 & 16 &-&-\\ 79 & 48 &4&-\\ \hline \end{array}

Thus, in that range, there are just $31$ unsolvables of form,

$$a^3+b^3+c^3 = (3m)^3$$

but a good $300$ unsolvables,

$$a^3+b^3+c^3 = (13m)^3$$

and $82$,

$$a^3+b^3+c^3 = (37m)^3$$

Q: What could explain this behavior of $p=7,13,37$ with respect to cubes?

It seems rather surprising that, given the Diophantine equation,

$$a^3+b^3+c^3 = n^3$$

then a good $\color{red}{99.8\%}$ of $n<1000000$ are solvable in positive integers $a,b,c$. (See the discussion in this MSE post.)

Oleg567 gave a list of the $1867$ unsolvable $n$ in that range. More than half were primes, while the divisibility by prime $p$ of the unsolvable composites are given by the table below. Oleg noticed a curious behavior:

\begin{array}{|l|c|c|c|c|} \hline p & \mbox{# divisible by } p & ...\; p^2 & ...\;p^3 \\ \hline 2 & \color{blue}{177} & 41 & 22\\ 3 & 31 &-&-\\ 5 & 53 &-&-\\ 7 & \color{blue}{218} & 52 & 11 \\ 11 & 43 &-&-\\ 13 & \color{blue}{300} & 46 & 4 \\ 17 & 47 &-&-\\ 23 & 31 &-&-\\ 31 & 34 &-&-\\ 37 & \color{blue}{82} &-&-\\ 43 & 39 &-&-\\ 47 & 29 &-&-\\ 59 & 24 &1&-\\ 61 & 13 &-&-\\ 73 & 16 &-&-\\ 79 & 48 &4&-\\ \hline \end{array}

Thus, in that range, there are just $31$ unsolvables of form,

$$a^3+b^3+c^3 = (3m)^3$$

but $218$ unsolvables,

$$a^3+b^3+c^3 = (7m)^3$$

and $300$,

$$a^3+b^3+c^3 = (13m)^3$$

Q: What could explain this "anomalous" behavior of $p=2^k,7^k,13^k$ for $k=1,2,3$ with respect to cubes?

It seems rather surprising that, given the Diophantine equation,

$$a^3+b^3+c^3 = n^3$$

then a good $\color{red}{99.8\%}$ of $n<1000000$ are solvable in positive integers $a,b,c$. (See the discussion in this MSE post.)

Oleg567 gave a list of the $1867$ unsolvable $n$ in that range. More than half were primes, while the divisibility by prime $p$ of the unsolvable composites are given by the table:

\begin{array}{|l|c|c|c|c|} \hline p & \mbox{# divisible by } p & ...\; p^2 & ...\;p^3 \\ \hline 2 & \color{blue}{177} & 41 & 22\\ 3 & 31 &-&-\\ 5 & 53 &-&-\\ 7 & \color{blue}{218} & 52 & 11 \\ 11 & 43 &-&-\\ 13 & \color{blue}{300} & 46 & 4 \\ 17 & 47 &-&-\\ 23 & 31 &-&-\\ 31 & 34 &-&-\\ 37 & \color{blue}{82} &-&-\\ 43 & 39 &-&-\\ 47 & 29 &-&-\\ 59 & 24 &1&-\\ 61 & 13 &-&-\\ 73 & 16 &-&-\\ 79 & 48 &4&-\\ \hline \end{array}

Thus, in that range, there are just $31$ unsolvables of form,

$$a^3+b^3+c^3 = (3m)^3$$

but a good $300$ unsolvables,

$$a^3+b^3+c^3 = (13m)^3$$

and $82$,

$$a^3+b^3+c^3 = (37m)^3$$

Q: What could explain this behavior of $p=7,13,37$ with respect to cubes?

It seems rather surprising that, given the Diophantine equation,

$$a^3+b^3+c^3 = n^3$$

then a good $\color{red}{99.8\%}$ of $n<1000000$ are solvable in positive integers $a,b,c$. (See the discussion in this MSE post.)

Oleg567 gave a list of the $1867$ unsolvable $n$ in that range. More than half were primes, while the divisibility by prime $p$ of the unsolvable composites are given by the table below. Oleg noticed a curious behavior:

\begin{array}{|l|c|c|c|c|} \hline p & \mbox{# divisible by } p & ...\; p^2 & ...\;p^3 \\ \hline 2 & \color{blue}{177} & 41 & 22\\ 3 & 31 &-&-\\ 5 & 53 &-&-\\ 7 & \color{blue}{218} & 52 & 11 \\ 11 & 43 &-&-\\ 13 & \color{blue}{300} & 46 & 4 \\ 17 & 47 &-&-\\ 23 & 31 &-&-\\ 31 & 34 &-&-\\ 37 & \color{blue}{82} &-&-\\ 43 & 39 &-&-\\ 47 & 29 &-&-\\ 59 & 24 &1&-\\ 61 & 13 &-&-\\ 73 & 16 &-&-\\ 79 & 48 &4&-\\ \hline \end{array}

Thus, in that range, there are just $31$ unsolvables of form,

$$a^3+b^3+c^3 = (3m)^3$$

but a good $300$ unsolvables,

$$a^3+b^3+c^3 = (13m)^3$$

and $82$,

$$a^3+b^3+c^3 = (37m)^3$$

Q: What could explain this behavior of $p=7,13,37$ with respect to cubes?