On $1^2+2^2+\dots+24^2 = 70^2$, and $15^3+16^3+\dots+34^3 = 70^3$

Just as a wide comment.

My search limitation was $x<2 \times 10^6$.

There is only $x=70$ in this range.

The set of such $x$, which can be written as sum of consequent cubes, is rather pure: here is list:

\begin{array}{rlr} 6^3 = & 3^3+4^3+5^3 & (3 ~terms) \\ 20^3 = & 11^3 + ... + 14^3 & (4= \color{darkviolet}{2^2}~terms) \\ 40^3 = & 3^3 + ... + 22^3 & (20 ~terms) \\ 60^3 = & 6^3 + ... + 30^3 & (25 = \color{darkviolet}{5^2}~terms) \\ 70^3 = & 15^3 + ... + 34^3 & (20 ~terms) \\ 180^3 = & 6^3 + ... + 69^3 & (64 = \color{darkviolet}{8^2}= \color{red}{4^3} ~terms) \\ 330^3 = & 11^3 + ... + 109^3 & (99 ~terms) \\ 540^3 = & 34^3 + ... + 158^3 & (125 = \color{red}{5^3} ~terms) \\ 1155^3 = & 291^3 + ... + 339^3 & (49 = \color{darkviolet}{7^2} ~terms) \\ 1581^3 = & 213^3 + ... + 365^3 & (153 ~terms) \\ 2805^3 = & 556^3 + ... + 654^3 & (99 ~terms) \\ 2856^3 = & 213^3 + ... + 555^3 & (343 = \color{red}{7^3} ~terms) \\ 2856^3 = & 273^3 + ... + 560^3 & (288 ~terms) \\ 3876^3 = & 646^3 + ... + 798^3 & (153 ~terms) \\ 5544^3 = & 406^3 + ... + 917^3 & (512 = \color{red}{8^3} ~terms) \\ 16830^3 = & 1134^3 + ... + 2133^3 & (1000 = \color{red}{10^3}~terms) \\ 27060^3 = & 1735^3 + ... + 3065^3 & (1331 = \color{red}{11^3}~terms) \\ 62244^3 = & 3606^3 + ... + 5802^3 & (2197 = \color{red}{13^3}~terms) \\ 82680^3 = & 305^3 + ... + 6895^3 & (6591 ~terms) \\ 90090^3 = & 4966^3 + ... + 7709^3 & (2744 = \color{red}{14^3}~terms) \\ 175440^3 = & 8790^3 + ... + 12885^3 & (4096 = \color{darkviolet}{64^2} = \color{red}{16^3}~terms) \\ 237456^3 = & 11368^3 + ... + 16280^3 & (4913 = \color{red}{17^3}~terms) \\ 249424^3 = & 1624^3 + ... + 15784^3 & (14161 = \color{darkviolet}{119^2}~terms) \\ 273819^3 = & 3010^3 + ... + 16932^3 & (13923 ~terms) \\ 413820^3 = & 18171^3 + ... + 25029^3 & (6859 = \color{red}{19^3}~terms) \\ 431548^3 = & 34228^3 + ... + 36076^3 & (1849 = \color{darkviolet}{43^2}~terms) \\ 534660^3 = & 22534^3 + ... + 30533^3 & (8000 = \color{red}{20^3}~terms) \\ 860706^3 = & 33558^3 + ... + 44205^3 & (10648 = \color{red}{22^3}~terms) \\ 1074744^3 = & 40381^3 + ... + 52547^3 & (12167 = \color{red}{23^3} ~terms) \\ 1205750^3 = & 18551^3 + ... + 51674^3 & (33124 = \color{darkviolet}{182^2}~terms) \\ 1306620^3 = & 4880^3 + ... + 54655^3 & (49776 ~terms) \\ 1630200^3 = & 57084^3 + ... + 72708^3 & (15625 = \color{darkviolet}{125^2} = \color{red}{25^3} ~terms) \\ 1764070^3 = & 46690^3 + ... + 71890^3 & (25201 ~terms) \\ 1962820^3 = & 11170^3 + ... + 74170^3 &(63001 = \color{darkviolet}{251^2}~terms) \\ 1983150^3 = & 67150^3 + ... + 84725^3 & (17576 = \color{red}{26^3} ~terms) \\ \end{array}

And a few curious identities of this search:

power $3$: $$ 215^3+...+555^3 = {\Large{2856^3}} = 273^3+...+ 560^3 $$

power $2$: $$ 7^2+...+39^2 = {\Large{143^2}} = 38^2+...+ 48^2 $$ $$ 294^2+...+367^2 = {\Large{2849^2}} = 854^2+...+ 864^2 $$ $$ 2175^2+...+5199^2 = {\Large{208395^2}} = 29447^2+...+ 29496^2 $$ $$ 9401^2+...+25273^2 = {\Large{2259257^2}} = 26181^2+...+ 32158^2 $$