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All integers except possibly those of the form $9k \pm 4$ can be written as the sum of 4 cubes.

Are there any heuristics that suggest this is impossible in general, or is it the case that no one has found any identities?

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  • $\begingroup$ According to the answer from Will Jagy that you cite, it is suspected that 4 cubes suffice in general, so it must be simply that it hasn't been proved. $\endgroup$ – saulspatz Apr 24 '18 at 16:55
  • $\begingroup$ Something you should know: $$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \text{Remainder of }a\div9 &0&1&2&3&4&5&6&7&8\ \\ \hline \text{Remainder of }a^3\div9 &0&1&8&0&1&8&0&1&8\\\hline \end{array}$$ $\endgroup$ – user061703 Apr 24 '18 at 17:13
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    $\begingroup$ $4=1^3+1^3+1^3+1^3$. $\endgroup$ – Théophile Apr 24 '18 at 17:17
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(Updated):

If the number is of the form $9k\pm 4$, then it cannot be written as the sum of three cubes. Simply because any cube number has form $9s$ or $9s\pm 1$.

As for four cubes: maybe it is not proven that all numbers of the form $9k\pm 4$ can be written as the sum of four cubes.


It looks that for many values $k$ it is relatively easy to decompose numbers of the form $9k\pm 4$ to the sum of $4$ cubes.

I checked $k$ in the range $1\le k \le 10^4$, and result is:
if $1\le k \le 10000$, then equation $$9k-4=a^3+b^3+c^3+d^4\tag{1}$$ has integer solution $(k,a,b,c,d)$, where $\max\{|a|,|b|,|c|,|d|\}<300$.
Few relatively "hard" examples from this set:
$9\cdot 3112-4 = 28004 = (-175)^3 + (-214)^3 + (-226)^3 + 299^3$;
$9\cdot 9174-4 = 82562 = 104^3 + (-196)^3 + (-235)^3 + 269^3$;
$9\cdot 1264-4 = 11372 = 59^3 + 59^3 + 257^3 + (-259)^3$;
$9\cdot 1418-4 = 12758 = (-58)^3 + (-169)^3 + (-217)^3 + 248^3$;
$9\cdot 3105-4 = 27941 = (-100)^3 + (-106)^3 + (-232)^3 + 245^3$;

and equation $$9k+4=a^3+b^3+c^3+d^4\tag{2}$$ has integer solution $(k,a,b,c,d)$, where $\max\{|a|,|b|,|c|,|d|\}<520$.
Few relatively "hard" examples from this set:
$9\cdot 2474+4 = 22270 = 325^3 + 349^3 + 391^3 + (-515)^3$;
$9\cdot 6517+4 = 58657 = (-161)^3 + (-221)^3 + (-401)^3 + 430^3$;
$9\cdot 6608+4 = 59476 = (-101)^3 + 247^3 + 295^3 + (-341)^3$;
$9\cdot 7479+4 = 67315 = (-221)^3 + 250^3 + 307^3 + (-323)^3$;
$9\cdot 574+4 = 5170 = 22^3 + 67^3 + 319^3 + (-320)^3$;


Here are few first numbers of the form $9k-4$:

 9*1-4 = 5 = (-1)^3 + (-1)^3 + (-1)^3 + 2^3;
 9*2-4 = 14 = (-1)^3 + (-1)^3 + 2^3 + 2^3;
 9*3-4 = 23 = (-1)^3 + 2^3 + 2^3 + 2^3;
 9*4-4 = 32 = 2^3 + 2^3 + 2^3 + 2^3;
 9*5-4 = 41 = (-4)^3 + (-4)^3 + (-7)^3 + 8^3;
 9*6-4 = 50 = 29^3 + 29^3 + 41^3 + (-49)^3;
 9*7-4 = 59 = (-1)^3 + (-1)^3 + (-4)^3 + 5^3;
 9*8-4 = 68 = (-1)^3 + 2^3 + (-4)^3 + 5^3;
 9*9-4 = 77 = 2^3 + 2^3 + (-4)^3 + 5^3;
 9*10-4 = 86 = 14^3 + 14^3 + 29^3 + (-31)^3;
 9*11-4 = 95 = (-1)^3 + 14^3 + 20^3 + (-22)^3;
 9*12-4 = 104 = (-1)^3 + (-4)^3 + (-7)^3 + 8^3;
 9*13-4 = 113 = 2^3 + (-4)^3 + (-7)^3 + 8^3;
 9*14-4 = 122 = (-1)^3 + (-1)^3 + (-1)^3 + 5^3;
 9*15-4 = 131 = (-1)^3 + (-1)^3 + 2^3 + 5^3;
 9*16-4 = 140 = (-1)^3 + 2^3 + 2^3 + 5^3;
 9*17-4 = 149 = 2^3 + 2^3 + 2^3 + 5^3;
 9*18-4 = 158 = 8^3 + 8^3 + 11^3 + (-13)^3;
 9*19-4 = 167 = (-1)^3 + (-1)^3 + (-7)^3 + 8^3;
 9*20-4 = 176 = (-1)^3 + 2^3 + (-7)^3 + 8^3;
 9*21-4 = 185 = (-1)^3 + (-4)^3 + 5^3 + 5^3;
 9*22-4 = 194 = 2^3 + (-4)^3 + 5^3 + 5^3;
 9*23-4 = 203 = (-4)^3 + (-4)^3 + (-10)^3 + 11^3;
 9*24-4 = 212 = 2^3 + (-7)^3 + (-13)^3 + 14^3;
 9*25-4 = 221 = 5^3 + 14^3 + 20^3 + (-22)^3;
 9*26-4 = 230 = (-4)^3 + 5^3 + (-7)^3 + 8^3;
 9*27-4 = 239 = (-1)^3 + (-13)^3 + (-28)^3 + 29^3;
 9*28-4 = 248 = (-1)^3 + (-1)^3 + 5^3 + 5^3;
 9*29-4 = 257 = (-1)^3 + 2^3 + 5^3 + 5^3;
 9*30-4 = 266 = 2^3 + 2^3 + 5^3 + 5^3;
 9*31-4 = 275 = 2^3 + (-4)^3 + (-10)^3 + 11^3;
 9*32-4 = 284 = 5^3 + 80^3 + 119^3 + (-130)^3;
 9*33-4 = 293 = (-1)^3 + 5^3 + (-7)^3 + 8^3;
 9*34-4 = 302 = 2^3 + 5^3 + (-7)^3 + 8^3;
 9*35-4 = 311 = (-4)^3 + 5^3 + 5^3 + 5^3;
 9*36-4 = 320 = (-4)^3 + (-4)^3 + (-4)^3 + 8^3;
 9*37-4 = 329 = (-1)^3 + (-1)^3 + (-10)^3 + 11^3;
 9*38-4 = 338 = (-7)^3 + (-7)^3 + 8^3 + 8^3;
 9*39-4 = 347 = 2^3 + 2^3 + (-10)^3 + 11^3;
 9*40-4 = 356 = (-7)^3 + (-22)^3 + (-61)^3 + 62^3;
 9*41-4 = 365 = 5^3 + (-13)^3 + (-28)^3 + 29^3;
 9*42-4 = 374 = (-1)^3 + 5^3 + 5^3 + 5^3;
 9*43-4 = 383 = 2^3 + 5^3 + 5^3 + 5^3;
 9*44-4 = 392 = 2^3 + (-4)^3 + (-4)^3 + 8^3;
 9*45-4 = 401 = (-4)^3 + 11^3 + 11^3 + (-13)^3;
 9*46-4 = 410 = (-4)^3 + (-7)^3 + (-16)^3 + 17^3;
 9*47-4 = 419 = 5^3 + 5^3 + (-7)^3 + 8^3;
 9*48-4 = 428 = 5^3 + 32^3 + 44^3 + (-49)^3;
 9*49-4 = 437 = (-10)^3 + (-10)^3 + (-28)^3 + 29^3;
 9*50-4 = 446 = (-1)^3 + (-1)^3 + (-4)^3 + 8^3;
 9*51-4 = 455 = (-1)^3 + 2^3 + (-4)^3 + 8^3;
 9*52-4 = 464 = 2^3 + 2^3 + (-4)^3 + 8^3;
 9*53-4 = 473 = 2^3 + 11^3 + 11^3 + (-13)^3;
 9*54-4 = 482 = (-1)^3 + (-4)^3 + (-13)^3 + 14^3;
 9*55-4 = 491 = 2^3 + (-4)^3 + (-13)^3 + 14^3;
 9*56-4 = 500 = 5^3 + 5^3 + 5^3 + 5^3;
 9*57-4 = 509 = (-1)^3 + (-1)^3 + (-1)^3 + 8^3;
 9*58-4 = 518 = (-1)^3 + (-1)^3 + 2^3 + 8^3;
 9*59-4 = 527 = (-1)^3 + 2^3 + 2^3 + 8^3;
 9*60-4 = 536 = 2^3 + 2^3 + 2^3 + 8^3;
 9*61-4 = 545 = (-1)^3 + (-1)^3 + (-13)^3 + 14^3;
 9*62-4 = 554 = (-1)^3 + 2^3 + (-13)^3 + 14^3;
 9*63-4 = 563 = 2^3 + 2^3 + (-13)^3 + 14^3;
 9*64-4 = 572 = (-1)^3 + (-4)^3 + 5^3 + 8^3;
 9*65-4 = 581 = 2^3 + (-4)^3 + 5^3 + 8^3;
 9*66-4 = 590 = 5^3 + 11^3 + 11^3 + (-13)^3;
 9*67-4 = 599 = 5^3 + (-7)^3 + (-16)^3 + 17^3;
 9*68-4 = 608 = (-4)^3 + 5^3 + (-13)^3 + 14^3;
 9*69-4 = 617 = (-4)^3 + (-7)^3 + 8^3 + 8^3;
 9*70-4 = 626 = 14^3 + 65^3 + 65^3 + (-82)^3;
 9*71-4 = 635 = (-1)^3 + (-1)^3 + 5^3 + 8^3;
 9*72-4 = 644 = (-1)^3 + 2^3 + 5^3 + 8^3;
 9*73-4 = 653 = 2^3 + 2^3 + 5^3 + 8^3;
 9*74-4 = 662 = (-10)^3 + (-10)^3 + 11^3 + 11^3;
 9*75-4 = 671 = (-1)^3 + 5^3 + (-13)^3 + 14^3;
 9*76-4 = 680 = (-1)^3 + (-7)^3 + 8^3 + 8^3;
 9*77-4 = 689 = 2^3 + (-7)^3 + 8^3 + 8^3;
 9*78-4 = 698 = (-4)^3 + 5^3 + 5^3 + 8^3;
 9*79-4 = 707 = (-10)^3 + (-13)^3 + (-16)^3 + 20^3;
 9*80-4 = 716 = (-7)^3 + 8^3 + (-13)^3 + 14^3;
 9*81-4 = 725 = (-43)^3 + 56^3 + 59^3 + (-67)^3;
 9*82-4 = 734 = (-4)^3 + 14^3 + 17^3 + (-19)^3;
 9*83-4 = 743 = (-1)^3 + (-10)^3 + (-10)^3 + 14^3;
 9*84-4 = 752 = 2^3 + (-10)^3 + (-10)^3 + 14^3;
 9*85-4 = 761 = (-1)^3 + 5^3 + 5^3 + 8^3;
 9*86-4 = 770 = 2^3 + 5^3 + 5^3 + 8^3;
 9*87-4 = 779 = (-4)^3 + 8^3 + (-10)^3 + 11^3;
 9*88-4 = 788 = 2^3 + (-13)^3 + (-31)^3 + 32^3;
 9*89-4 = 797 = 5^3 + 5^3 + (-13)^3 + 14^3;
 9*90-4 = 806 = 5^3 + (-7)^3 + 8^3 + 8^3;
 9*91-4 = 815 = (-1)^3 + (-1)^3 + (-16)^3 + 17^3;
 9*92-4 = 824 = (-1)^3 + 2^3 + (-16)^3 + 17^3;
 9*93-4 = 833 = 2^3 + 2^3 + (-16)^3 + 17^3;
 9*94-4 = 842 = (-1)^3 + 8^3 + (-10)^3 + 11^3;
 9*95-4 = 851 = 2^3 + 8^3 + (-10)^3 + 11^3;
 9*96-4 = 860 = (-4)^3 + (-4)^3 + (-7)^3 + 11^3;
 9*97-4 = 869 = 5^3 + (-10)^3 + (-10)^3 + 14^3;
 9*98-4 = 878 = (-10)^3 + 11^3 + (-13)^3 + 14^3;
 9*99-4 = 887 = 5^3 + 5^3 + 5^3 + 8^3;
 9*100-4 = 896 = (-4)^3 + (-4)^3 + 8^3 + 8^3;
 9*101-4 = 905 = (-10)^3 + (-19)^3 + (-25)^3 + 29^3;
 9*102-4 = 914 = (-13)^3 + (-13)^3 + (-19)^3 + 23^3;
 9*103-4 = 923 = (-1)^3 + (-4)^3 + (-7)^3 + 11^3;
 9*104-4 = 932 = 2^3 + (-4)^3 + (-7)^3 + 11^3;
 9*105-4 = 941 = (-1)^3 + 5^3 + (-16)^3 + 17^3;
 9*106-4 = 950 = 2^3 + 5^3 + (-16)^3 + 17^3;
 9*107-4 = 959 = (-1)^3 + (-4)^3 + 8^3 + 8^3;
 9*108-4 = 968 = 2^3 + (-4)^3 + 8^3 + 8^3;
 9*109-4 = 977 = 8^3 + 11^3 + 11^3 + (-13)^3;
 9*110-4 = 986 = (-1)^3 + (-1)^3 + (-7)^3 + 11^3;
 9*111-4 = 995 = (-1)^3 + 2^3 + (-7)^3 + 11^3;
 9*112-4 = 1004 = 2^3 + 2^3 + (-7)^3 + 11^3;
 9*113-4 = 1013 = (-4)^3 + (-4)^3 + (-19)^3 + 20^3;
 9*114-4 = 1022 = (-1)^3 + (-1)^3 + 8^3 + 8^3;
 9*115-4 = 1031 = (-1)^3 + 2^3 + 8^3 + 8^3;
 9*116-4 = 1040 = 2^3 + 2^3 + 8^3 + 8^3;
 9*117-4 = 1049 = (-4)^3 + 5^3 + (-7)^3 + 11^3;
 9*118-4 = 1058 = (-7)^3 + (-7)^3 + (-10)^3 + 14^3;
 9*119-4 = 1067 = 2^3 + 8^3 + (-13)^3 + 14^3;
 9*120-4 = 1076 = (-1)^3 + (-4)^3 + (-19)^3 + 20^3;
 9*121-4 = 1085 = (-4)^3 + 5^3 + 8^3 + 8^3;
 9*122-4 = 1094 = (-13)^3 + (-13)^3 + 14^3 + 14^3;
 9*123-4 = 1103 = 38^3 + 47^3 + 77^3 + (-85)^3;
 9*124-4 = 1112 = (-1)^3 + 5^3 + (-7)^3 + 11^3;
 9*125-4 = 1121 = 2^3 + 5^3 + (-7)^3 + 11^3;
 9*126-4 = 1130 = 29^3 + (-91)^3 + (-94)^3 + 116^3;
 9*127-4 = 1139 = (-4)^3 + (-4)^3 + (-4)^3 + 11^3;
 9*128-4 = 1148 = (-1)^3 + 5^3 + 8^3 + 8^3;
 9*129-4 = 1157 = 2^3 + 5^3 + 8^3 + 8^3;
 9*130-4 = 1166 = 41^3 + (-64)^3 + (-64)^3 + 77^3;
 9*131-4 = 1175 = (-1)^3 + (-7)^3 + (-22)^3 + 23^3;
 9*132-4 = 1184 = 5^3 + 8^3 + (-13)^3 + 14^3;
 9*133-4 = 1193 = (-7)^3 + 8^3 + 8^3 + 8^3;
 9*134-4 = 1202 = (-1)^3 + (-4)^3 + (-4)^3 + 11^3;
 9*135-4 = 1211 = 2^3 + (-4)^3 + (-4)^3 + 11^3;
 9*136-4 = 1220 = 2^3 + (-16)^3 + (-19)^3 + 23^3;
 9*137-4 = 1229 = 17^3 + (-28)^3 + (-37)^3 + 41^3;
 9*138-4 = 1238 = 5^3 + 5^3 + (-7)^3 + 11^3;
 9*139-4 = 1247 = (-52)^3 + (-94)^3 + (-121)^3 + 140^3;
 9*140-4 = 1256 = 8^3 + (-10)^3 + (-10)^3 + 14^3;
 9*141-4 = 1265 = (-1)^3 + (-1)^3 + (-4)^3 + 11^3;
 9*142-4 = 1274 = 5^3 + 5^3 + 8^3 + 8^3;
 9*143-4 = 1283 = 2^3 + 2^3 + (-4)^3 + 11^3;
 9*144-4 = 1292 = 8^3 + (-13)^3 + (-31)^3 + 32^3;
 9*145-4 = 1301 = 5^3 + (-7)^3 + (-22)^3 + 23^3;
 9*146-4 = 1310 = 11^3 + 11^3 + 14^3 + (-16)^3;
 9*147-4 = 1319 = (-7)^3 + (-10)^3 + 11^3 + 11^3;
 9*148-4 = 1328 = (-1)^3 + (-1)^3 + (-1)^3 + 11^3;
 9*149-4 = 1337 = (-1)^3 + (-1)^3 + 2^3 + 11^3;
 9*150-4 = 1346 = (-1)^3 + 2^3 + 2^3 + 11^3;
 9*151-4 = 1355 = 2^3 + 2^3 + 2^3 + 11^3;
 9*152-4 = 1364 = (-13)^3 + 14^3 + (-16)^3 + 17^3;
 9*153-4 = 1373 = (-7)^3 + (-10)^3 + (-13)^3 + 17^3;
 9*154-4 = 1382 = (-10)^3 + 23^3 + 23^3 + (-28)^3;
 9*155-4 = 1391 = (-1)^3 + (-4)^3 + 5^3 + 11^3;
 9*156-4 = 1400 = 2^3 + (-4)^3 + 5^3 + 11^3;
 9*157-4 = 1409 = 2^3 + (-7)^3 + (-10)^3 + 14^3;
 9*158-4 = 1418 = (-31)^3 + (-31)^3 + (-40)^3 + 50^3;
 9*159-4 = 1427 = 11^3 + 14^3 + 20^3 + (-22)^3;
 9*160-4 = 1436 = (-4)^3 + (-7)^3 + 8^3 + 11^3;
 9*161-4 = 1445 = 2^3 + (-10)^3 + (-28)^3 + 29^3;
 9*162-4 = 1454 = (-1)^3 + (-1)^3 + 5^3 + 11^3;
 9*163-4 = 1463 = (-1)^3 + 2^3 + 5^3 + 11^3;
 9*164-4 = 1472 = (-4)^3 + 8^3 + 8^3 + 8^3;
 9*165-4 = 1481 = (-19)^3 + (-34)^3 + (-61)^3 + 65^3;
 9*166-4 = 1490 = (-7)^3 + (-22)^3 + (-64)^3 + 65^3;
 9*167-4 = 1499 = (-1)^3 + (-7)^3 + 8^3 + 11^3;
 9*168-4 = 1508 = 2^3 + (-7)^3 + 8^3 + 11^3;
 9*169-4 = 1517 = (-4)^3 + 5^3 + 5^3 + 11^3;
 9*170-4 = 1526 = 5^3 + (-7)^3 + (-10)^3 + 14^3;
 9*171-4 = 1535 = (-1)^3 + 8^3 + 8^3 + 8^3;
 9*172-4 = 1544 = 2^3 + 8^3 + 8^3 + 8^3;
 9*173-4 = 1553 = 11^3 + 20^3 + 35^3 + (-37)^3;
 9*174-4 = 1562 = (-7)^3 + (-19)^3 + (-25)^3 + 29^3;
 9*175-4 = 1571 = 8^3 + 8^3 + (-13)^3 + 14^3;
 9*176-4 = 1580 = (-1)^3 + 5^3 + 5^3 + 11^3;
 9*177-4 = 1589 = 2^3 + 5^3 + 5^3 + 11^3;
 9*178-4 = 1598 = (-4)^3 + (-10)^3 + 11^3 + 11^3;
 9*179-4 = 1607 = (-1)^3 + (-7)^3 + (-25)^3 + 26^3;
 9*180-4 = 1616 = (-4)^3 + (-4)^3 + (-10)^3 + 14^3;
 9*181-4 = 1625 = 5^3 + (-7)^3 + 8^3 + 11^3;
 9*182-4 = 1634 = (-16)^3 + (-16)^3 + 17^3 + 17^3;
 9*183-4 = 1643 = (-4)^3 + (-13)^3 + (-16)^3 + 20^3;
 9*184-4 = 1652 = (-4)^3 + (-10)^3 + (-13)^3 + 17^3;
 9*185-4 = 1661 = 5^3 + 8^3 + 8^3 + 8^3;
 9*186-4 = 1670 = 2^3 + (-10)^3 + 11^3 + 11^3;
 9*187-4 = 1679 = (-1)^3 + (-4)^3 + (-10)^3 + 14^3;
 9*188-4 = 1688 = 2^3 + (-4)^3 + (-10)^3 + 14^3;
 9*189-4 = 1697 = (-1)^3 + 47^3 + 80^3 + (-85)^3;
 9*190-4 = 1706 = 5^3 + 5^3 + 5^3 + 11^3;
 9*191-4 = 1715 = (-4)^3 + (-4)^3 + 8^3 + 11^3;
 9*192-4 = 1724 = 2^3 + (-10)^3 + (-13)^3 + 17^3;
 9*193-4 = 1733 = 5^3 + (-7)^3 + (-25)^3 + 26^3;
 9*194-4 = 1742 = (-1)^3 + (-1)^3 + (-10)^3 + 14^3;
 9*195-4 = 1751 = (-1)^3 + 2^3 + (-10)^3 + 14^3;
 9*196-4 = 1760 = 2^3 + 2^3 + (-10)^3 + 14^3;
 9*197-4 = 1769 = 5^3 + 5^3 + (-22)^3 + 23^3;
 9*198-4 = 1778 = (-1)^3 + (-4)^3 + 8^3 + 11^3;
 9*199-4 = 1787 = 2^3 + (-4)^3 + 8^3 + 11^3;
 9*200-4 = 1796 = 11^3 + 11^3 + 11^3 + (-13)^3;
 9*201-4 = 1805 = (-4)^3 + 5^3 + (-10)^3 + 14^3;
 9*202-4 = 1814 = (-4)^3 + 11^3 + (-13)^3 + 14^3;
 9*203-4 = 1823 = (-4)^3 + (-4)^3 + (-25)^3 + 26^3;
 9*204-4 = 1832 = 5^3 + (-13)^3 + (-16)^3 + 20^3;
 9*205-4 = 1841 = (-1)^3 + (-1)^3 + 8^3 + 11^3;
 9*206-4 = 1850 = (-1)^3 + 2^3 + 8^3 + 11^3;
 9*207-4 = 1859 = 2^3 + 2^3 + 8^3 + 11^3;
 9*208-4 = 1868 = (-1)^3 + 5^3 + (-10)^3 + 14^3;
 9*209-4 = 1877 = 2^3 + 5^3 + (-10)^3 + 14^3;
 9*210-4 = 1886 = 2^3 + 11^3 + (-13)^3 + 14^3;
 9*211-4 = 1895 = 2^3 + (-4)^3 + (-25)^3 + 26^3;
 9*212-4 = 1904 = (-4)^3 + 5^3 + 8^3 + 11^3;
 9*213-4 = 1913 = (-7)^3 + 8^3 + (-10)^3 + 14^3;
 9*214-4 = 1922 = 14^3 + 17^3 + 17^3 + (-22)^3;
 9*215-4 = 1931 = (-22)^3 + (-28)^3 + (-37)^3 + 44^3;
 9*216-4 = 1940 = 2^3 + (-25)^3 + (-76)^3 + 77^3;
 9*217-4 = 1949 = (-1)^3 + (-1)^3 + (-25)^3 + 26^3;
 9*218-4 = 1958 = (-16)^3 + 17^3 + (-19)^3 + 20^3;
 9*219-4 = 1967 = (-1)^3 + 5^3 + 8^3 + 11^3;
 9*220-4 = 1976 = 2^3 + 5^3 + 8^3 + 11^3;
 9*221-4 = 1985 = 2^3 + (-10)^3 + (-31)^3 + 32^3;
 9*222-4 = 1994 = (-4)^3 + (-7)^3 + (-7)^3 + 14^3;
 9*223-4 = 2003 = 5^3 + 11^3 + (-13)^3 + 14^3;
 9*224-4 = 2012 = (-7)^3 + 8^3 + 8^3 + 11^3;
 9*225-4 = 2021 = (-1)^3 + (-13)^3 + (-37)^3 + 38^3;
 9*226-4 = 2030 = (-7)^3 + (-7)^3 + (-13)^3 + 17^3;
 9*227-4 = 2039 = 2^3 + 8^3 + (-22)^3 + 23^3;
 9*228-4 = 2048 = 8^3 + 8^3 + 8^3 + 8^3;
 9*229-4 = 2057 = (-1)^3 + (-7)^3 + (-7)^3 + 14^3;
 9*230-4 = 2066 = 2^3 + (-7)^3 + (-7)^3 + 14^3;
 9*231-4 = 2075 = (-10)^3 + (-10)^3 + 11^3 + 14^3;
 9*232-4 = 2084 = (-4)^3 + 11^3 + (-16)^3 + 17^3;
 9*233-4 = 2093 = 5^3 + 5^3 + 8^3 + 11^3;
 9*234-4 = 2102 = 2^3 + (-7)^3 + (-28)^3 + 29^3;
 9*235-4 = 2111 = (-10)^3 + (-13)^3 + (-19)^3 + 23^3;
 9*236-4 = 2120 = (-7)^3 + 8^3 + (-25)^3 + 26^3;
 9*237-4 = 2129 = 11^3 + 14^3 + 17^3 + (-19)^3;
 9*238-4 = 2138 = 20^3 + 47^3 + 83^3 + (-88)^3;
 9*239-4 = 2147 = (-1)^3 + 11^3 + (-16)^3 + 17^3;
 9*240-4 = 2156 = 2^3 + 11^3 + (-16)^3 + 17^3;
 9*241-4 = 2165 = 8^3 + 8^3 + (-19)^3 + 20^3;
 9*242-4 = 2174 = 8^3 + (-10)^3 + 11^3 + 11^3;
 9*243-4 = 2183 = 5^3 + (-7)^3 + (-7)^3 + 14^3;
 9*244-4 = 2192 = (-4)^3 + 8^3 + (-10)^3 + 14^3;
 9*245-4 = 2201 = (-4)^3 + 17^3 + 20^3 + (-22)^3;
 9*246-4 = 2210 = 8^3 + 47^3 + 80^3 + (-85)^3;
 9*247-4 = 2219 = 8^3 + (-13)^3 + (-16)^3 + 20^3;
 9*248-4 = 2228 = 8^3 + (-10)^3 + (-13)^3 + 17^3;
 9*249-4 = 2237 = (-160)^3 + 176^3 + 197^3 + (-208)^3;
 9*250-4 = 2246 = (-25)^3 + (-28)^3 + (-40)^3 + 47^3;
 9*251-4 = 2255 = (-4)^3 + (-7)^3 + 11^3 + 11^3;
 9*252-4 = 2264 = 2^3 + 8^3 + (-10)^3 + 14^3;
 9*253-4 = 2273 = (-4)^3 + (-4)^3 + (-7)^3 + 14^3;
 9*254-4 = 2282 = (-19)^3 + (-19)^3 + 20^3 + 20^3;
 9*255-4 = 2291 = (-4)^3 + 8^3 + 8^3 + 11^3;
 9*256-4 = 2300 = (-7)^3 + (-25)^3 + (-37)^3 + 41^3;
 9*257-4 = 2309 = (-4)^3 + (-7)^3 + (-13)^3 + 17^3;
 9*258-4 = 2318 = (-1)^3 + (-7)^3 + 11^3 + 11^3;
 9*259-4 = 2327 = 2^3 + (-7)^3 + 11^3 + 11^3;
 9*260-4 = 2336 = (-1)^3 + (-4)^3 + (-7)^3 + 14^3;
 9*261-4 = 2345 = 2^3 + (-4)^3 + (-7)^3 + 14^3;
 9*262-4 = 2354 = (-1)^3 + 8^3 + 8^3 + 11^3;
 9*263-4 = 2363 = 2^3 + 8^3 + 8^3 + 11^3;
 9*264-4 = 2372 = (-1)^3 + (-7)^3 + (-13)^3 + 17^3;
 9*265-4 = 2381 = 5^3 + 8^3 + (-10)^3 + 14^3;
 9*266-4 = 2390 = 8^3 + 11^3 + (-13)^3 + 14^3;
 9*267-4 = 2399 = (-1)^3 + (-1)^3 + (-7)^3 + 14^3;
 9*268-4 = 2408 = (-1)^3 + 2^3 + (-7)^3 + 14^3;
 9*269-4 = 2417 = 2^3 + 2^3 + (-7)^3 + 14^3;
 9*270-4 = 2426 = (-7)^3 + 41^3 + 104^3 + (-106)^3;
 ...

And few first numbers of the form $9k+4$:

 9*0+4 = 4 = 1^3 + 1^3 + 1^3 + 1^3;
 9*1+4 = 13 = 1^3 + 7^3 + 10^3 + (-11)^3;
 9*2+4 = 22 = 1^3 + (-11)^3 + (-14)^3 + 16^3;
 9*3+4 = 31 = 31^3 + (-44)^3 + (-44)^3 + 52^3;
 9*4+4 = 40 = (-2)^3 + (-2)^3 + (-2)^3 + 4^3;
 9*5+4 = 49 = 1^3 + (-2)^3 + (-2)^3 + 4^3;
 9*6+4 = 58 = 1^3 + 1^3 + (-2)^3 + 4^3;
 9*7+4 = 67 = 1^3 + 1^3 + 1^3 + 4^3;
 9*8+4 = 76 = 4^3 + 7^3 + 10^3 + (-11)^3;
 9*9+4 = 85 = (-2)^3 + (-5)^3 + (-5)^3 + 7^3;
 9*10+4 = 94 = 1^3 + (-5)^3 + (-5)^3 + 7^3;
 9*11+4 = 103 = (-11)^3 + (-11)^3 + (-11)^3 + 16^3;
 9*12+4 = 112 = (-2)^3 + (-2)^3 + 4^3 + 4^3;
 9*13+4 = 121 = 1^3 + (-2)^3 + 4^3 + 4^3;
 9*14+4 = 130 = 1^3 + 1^3 + 4^3 + 4^3;
 9*15+4 = 139 = 7^3 + 7^3 + 13^3 + (-14)^3;
 9*16+4 = 148 = 1^3 + (-50)^3 + (-56)^3 + 67^3;
 9*17+4 = 157 = 4^3 + (-5)^3 + (-5)^3 + 7^3;
 9*18+4 = 166 = (-2)^3 + 7^3 + 7^3 + (-8)^3;
 9*19+4 = 175 = 1^3 + 7^3 + 7^3 + (-8)^3;
 9*20+4 = 184 = (-2)^3 + 4^3 + 4^3 + 4^3;
 9*21+4 = 193 = 1^3 + 4^3 + 4^3 + 4^3;
 9*22+4 = 202 = (-2)^3 + (-2)^3 + (-5)^3 + 7^3;
 9*23+4 = 211 = 1^3 + (-2)^3 + (-5)^3 + 7^3;
 9*24+4 = 220 = 1^3 + 1^3 + (-5)^3 + 7^3;
 9*25+4 = 229 = (-5)^3 + (-8)^3 + (-11)^3 + 13^3;
 9*26+4 = 238 = 4^3 + 7^3 + 7^3 + (-8)^3;
 9*27+4 = 247 = 4^3 + 10^3 + 16^3 + (-17)^3;
 9*28+4 = 256 = 4^3 + 4^3 + 4^3 + 4^3;
 9*29+4 = 265 = 7^3 + (-26)^3 + (-53)^3 + 55^3;
 9*30+4 = 274 = (-2)^3 + 4^3 + (-5)^3 + 7^3;
 9*31+4 = 283 = 1^3 + 4^3 + (-5)^3 + 7^3;
 9*32+4 = 292 = 13^3 + 19^3 + 25^3 + (-29)^3;
 9*33+4 = 301 = (-11)^3 + (-14)^3 + (-26)^3 + 28^3;
 9*34+4 = 310 = (-8)^3 + (-17)^3 + (-17)^3 + 22^3;
 9*35+4 = 319 = (-2)^3 + (-2)^3 + (-2)^3 + 7^3;
 9*36+4 = 328 = 1^3 + (-2)^3 + (-2)^3 + 7^3;
 9*37+4 = 337 = 1^3 + 1^3 + (-2)^3 + 7^3;
 9*38+4 = 346 = 1^3 + 1^3 + 1^3 + 7^3;
 9*39+4 = 355 = (-2)^3 + (-5)^3 + (-8)^3 + 10^3;
 9*40+4 = 364 = 1^3 + (-5)^3 + (-8)^3 + 10^3;
 9*41+4 = 373 = 22^3 + 22^3 + 28^3 + (-35)^3;
 9*42+4 = 382 = (-23)^3 + (-35)^3 + (-44)^3 + 52^3;
 9*43+4 = 391 = (-2)^3 + (-2)^3 + 4^3 + 7^3;
 9*44+4 = 400 = 1^3 + (-2)^3 + 4^3 + 7^3;
 9*45+4 = 409 = 1^3 + 1^3 + 4^3 + 7^3;
 9*46+4 = 418 = 4^3 + (-8)^3 + (-11)^3 + 13^3;
 9*47+4 = 427 = 4^3 + (-5)^3 + (-8)^3 + 10^3;
 9*48+4 = 436 = (-5)^3 + (-5)^3 + 7^3 + 7^3;
 9*49+4 = 445 = (-2)^3 + 10^3 + 13^3 + (-14)^3;
 9*50+4 = 454 = 1^3 + 10^3 + 13^3 + (-14)^3;
 9*51+4 = 463 = (-2)^3 + 4^3 + 4^3 + 7^3;
 9*52+4 = 472 = 1^3 + 4^3 + 4^3 + 7^3;
 9*53+4 = 481 = 1^3 + (-2)^3 + (-8)^3 + 10^3;
 9*54+4 = 490 = 1^3 + 1^3 + (-8)^3 + 10^3;
 9*55+4 = 499 = (-26)^3 + (-29)^3 + (-38)^3 + 46^3;
 9*56+4 = 508 = 10^3 + 19^3 + 49^3 + (-50)^3;
 9*57+4 = 517 = 7^3 + 7^3 + 7^3 + (-8)^3;
 9*58+4 = 526 = 7^3 + 10^3 + 16^3 + (-17)^3;
 9*59+4 = 535 = 4^3 + 4^3 + 4^3 + 7^3;
 9*60+4 = 544 = (-2)^3 + 4^3 + (-8)^3 + 10^3;
 9*61+4 = 553 = (-2)^3 + (-5)^3 + 7^3 + 7^3;
 9*62+4 = 562 = 1^3 + (-5)^3 + 7^3 + 7^3;
 9*63+4 = 571 = 40^3 + 91^3 + 160^3 + (-170)^3;
 9*64+4 = 580 = 16^3 + 40^3 + 73^3 + (-77)^3;
 9*65+4 = 589 = (-5)^3 + (-14)^3 + (-23)^3 + 25^3;
 9*66+4 = 598 = (-5)^3 + 58^3 + 64^3 + (-77)^3;
 9*67+4 = 607 = (-2)^3 + (-11)^3 + (-17)^3 + 19^3;
 9*68+4 = 616 = 4^3 + 4^3 + (-8)^3 + 10^3;
 9*69+4 = 625 = 4^3 + (-5)^3 + 7^3 + 7^3;
 9*70+4 = 634 = 19^3 + 19^3 + 31^3 + (-35)^3;
 9*71+4 = 643 = (-5)^3 + 28^3 + 40^3 + (-44)^3;
 9*72+4 = 652 = 7^3 + (-26)^3 + (-32)^3 + 37^3;
 9*73+4 = 661 = (-2)^3 + 10^3 + 10^3 + (-11)^3;
 9*74+4 = 670 = (-2)^3 + (-2)^3 + 7^3 + 7^3;
 9*75+4 = 679 = 1^3 + (-2)^3 + 7^3 + 7^3;
 9*76+4 = 688 = 1^3 + 1^3 + 7^3 + 7^3;
 9*77+4 = 697 = 7^3 + (-8)^3 + (-11)^3 + 13^3;
 9*78+4 = 706 = (-5)^3 + 7^3 + (-8)^3 + 10^3;
 9*79+4 = 715 = (-5)^3 + (-8)^3 + (-14)^3 + 16^3;
 9*80+4 = 724 = 7^3 + 22^3 + 58^3 + (-59)^3;
 9*81+4 = 733 = 4^3 + 10^3 + 10^3 + (-11)^3;
 9*82+4 = 742 = (-2)^3 + 4^3 + 7^3 + 7^3;
 9*83+4 = 751 = 1^3 + 4^3 + 7^3 + 7^3;
 9*84+4 = 760 = 10^3 + 13^3 + 28^3 + (-29)^3;
 9*85+4 = 769 = 1^3 + 28^3 + 40^3 + (-44)^3;
 9*86+4 = 778 = 4^3 + (-14)^3 + (-23)^3 + 25^3;
 9*87+4 = 787 = 13^3 + 19^3 + 52^3 + (-53)^3;
 9*88+4 = 796 = 7^3 + 10^3 + 13^3 + (-14)^3;
 9*89+4 = 805 = 4^3 + (-5)^3 + (-11)^3 + 13^3;
 9*90+4 = 814 = 4^3 + 4^3 + 7^3 + 7^3;
 9*91+4 = 823 = (-2)^3 + 7^3 + (-8)^3 + 10^3;
 9*92+4 = 832 = 1^3 + 7^3 + (-8)^3 + 10^3;
 9*93+4 = 841 = 1^3 + (-8)^3 + (-14)^3 + 16^3;
 9*94+4 = 850 = (-2)^3 + (-2)^3 + (-11)^3 + 13^3;
 9*95+4 = 859 = (-2)^3 + (-2)^3 + (-5)^3 + 10^3;
 9*96+4 = 868 = 1^3 + (-2)^3 + (-5)^3 + 10^3;
 9*97+4 = 877 = 1^3 + 1^3 + (-5)^3 + 10^3;
 9*98+4 = 886 = 4^3 + (-17)^3 + (-17)^3 + 22^3;
 9*99+4 = 895 = 4^3 + 7^3 + (-8)^3 + 10^3;
 9*100+4 = 904 = (-5)^3 + 7^3 + 7^3 + 7^3;
 9*101+4 = 913 = (-17)^3 + (-20)^3 + (-47)^3 + 49^3;
 9*102+4 = 922 = (-2)^3 + 4^3 + (-11)^3 + 13^3;
 9*103+4 = 931 = (-2)^3 + 4^3 + (-5)^3 + 10^3;
 9*104+4 = 940 = 1^3 + 4^3 + (-5)^3 + 10^3;
 9*105+4 = 949 = (-17)^3 + (-47)^3 + (-83)^3 + 88^3;
 9*106+4 = 958 = 7^3 + (-11)^3 + (-17)^3 + 19^3;
 9*107+4 = 967 = 16^3 + 28^3 + 31^3 + (-38)^3;
 9*108+4 = 976 = (-2)^3 + (-2)^3 + (-2)^3 + 10^3;
 9*109+4 = 985 = 1^3 + (-2)^3 + (-2)^3 + 10^3;
 9*110+4 = 994 = 1^3 + 1^3 + (-2)^3 + 10^3;
 9*111+4 = 1003 = 1^3 + 1^3 + 1^3 + 10^3;
 9*112+4 = 1012 = 7^3 + 10^3 + 10^3 + (-11)^3;
 9*113+4 = 1021 = (-2)^3 + 7^3 + 7^3 + 7^3;
 9*114+4 = 1030 = 1^3 + 7^3 + 7^3 + 7^3;
 9*115+4 = 1039 = (-8)^3 + 19^3 + 19^3 + (-23)^3;
 9*116+4 = 1048 = (-2)^3 + (-2)^3 + 4^3 + 10^3;
 9*117+4 = 1057 = 1^3 + (-2)^3 + 4^3 + 10^3;
 9*118+4 = 1066 = 1^3 + 1^3 + 4^3 + 10^3;
 9*119+4 = 1075 = 28^3 + 40^3 + 40^3 + (-53)^3;
 9*120+4 = 1084 = (-5)^3 + 7^3 + (-11)^3 + 13^3;
 9*121+4 = 1093 = 4^3 + 7^3 + 7^3 + 7^3;
 9*122+4 = 1102 = (-5)^3 + (-5)^3 + (-14)^3 + 16^3;
 9*123+4 = 1111 = (-2)^3 + 16^3 + 31^3 + (-32)^3;
 9*124+4 = 1120 = (-2)^3 + 4^3 + 4^3 + 10^3;
 9*125+4 = 1129 = 1^3 + 4^3 + 4^3 + 10^3;
 9*126+4 = 1138 = (-8)^3 + 13^3 + 13^3 + (-14)^3;
 9*127+4 = 1147 = (-8)^3 + 16^3 + 28^3 + (-29)^3;
 9*128+4 = 1156 = 22^3 + 25^3 + 91^3 + (-92)^3;
 9*129+4 = 1165 = (-2)^3 + (-8)^3 + (-8)^3 + 13^3;
 9*130+4 = 1174 = 7^3 + 7^3 + (-8)^3 + 10^3;
 9*131+4 = 1183 = 7^3 + (-8)^3 + (-14)^3 + 16^3;
 9*132+4 = 1192 = 4^3 + 4^3 + 4^3 + 10^3;
 9*133+4 = 1201 = (-2)^3 + 7^3 + (-11)^3 + 13^3;
 9*134+4 = 1210 = (-2)^3 + (-5)^3 + 7^3 + 10^3;
 9*135+4 = 1219 = 1^3 + (-5)^3 + 7^3 + 10^3;
 9*136+4 = 1228 = 1^3 + (-5)^3 + (-14)^3 + 16^3;
 9*137+4 = 1237 = 4^3 + (-8)^3 + (-8)^3 + 13^3;
 9*138+4 = 1246 = (-5)^3 + (-14)^3 + (-14)^3 + 19^3;
 9*139+4 = 1255 = (-5)^3 + 13^3 + 16^3 + (-17)^3;
 9*140+4 = 1264 = 16^3 + 16^3 + 22^3 + (-26)^3;
 9*141+4 = 1273 = 4^3 + 7^3 + (-11)^3 + 13^3;
 9*142+4 = 1282 = 4^3 + (-5)^3 + 7^3 + 10^3;
 9*143+4 = 1291 = 4^3 + (-5)^3 + (-14)^3 + 16^3;
 9*144+4 = 1300 = 19^3 + 25^3 + 40^3 + (-44)^3;
 9*145+4 = 1309 = (-5)^3 + (-11)^3 + (-11)^3 + 16^3;
 9*146+4 = 1318 = 1^3 + (-11)^3 + (-20)^3 + 22^3;
 9*147+4 = 1327 = (-2)^3 + (-2)^3 + 7^3 + 10^3;
 9*148+4 = 1336 = 1^3 + (-2)^3 + 7^3 + 10^3;
 9*149+4 = 1345 = 1^3 + 1^3 + 7^3 + 10^3;
 9*150+4 = 1354 = (-8)^3 + 10^3 + (-11)^3 + 13^3;
 9*151+4 = 1363 = (-5)^3 + (-8)^3 + 10^3 + 10^3;
 9*152+4 = 1372 = 7^3 + 7^3 + 7^3 + 7^3;
 9*153+4 = 1381 = 1^3 + 13^3 + 16^3 + (-17)^3;
 9*154+4 = 1390 = 13^3 + 73^3 + 82^3 + (-98)^3;
 9*155+4 = 1399 = (-2)^3 + 4^3 + 7^3 + 10^3;
 9*156+4 = 1408 = 1^3 + 4^3 + 7^3 + 10^3;
 9*157+4 = 1417 = 1^3 + 4^3 + (-14)^3 + 16^3;
 9*158+4 = 1426 = (-2)^3 + (-11)^3 + (-11)^3 + 16^3;
 9*159+4 = 1435 = (-5)^3 + (-5)^3 + (-8)^3 + 13^3;
 9*160+4 = 1444 = 4^3 + 13^3 + 16^3 + (-17)^3;
 9*161+4 = 1453 = 10^3 + 10^3 + 13^3 + (-14)^3;
 9*162+4 = 1462 = 7^3 + 16^3 + 31^3 + (-32)^3;
 9*163+4 = 1471 = 4^3 + 4^3 + 7^3 + 10^3;
 9*164+4 = 1480 = (-2)^3 + (-8)^3 + 10^3 + 10^3;
 9*165+4 = 1489 = 1^3 + (-8)^3 + 10^3 + 10^3;
 9*166+4 = 1498 = 4^3 + (-11)^3 + (-11)^3 + 16^3;
 9*167+4 = 1507 = 19^3 + (-20)^3 + (-20)^3 + 22^3;
 9*168+4 = 1516 = 7^3 + (-8)^3 + (-8)^3 + 13^3;
 9*169+4 = 1525 = (-5)^3 + 13^3 + 13^3 + (-14)^3;
 9*170+4 = 1534 = (-5)^3 + 16^3 + 28^3 + (-29)^3;
 9*171+4 = 1543 = (-2)^3 + 19^3 + 19^3 + (-23)^3;
 9*172+4 = 1552 = 4^3 + (-8)^3 + 10^3 + 10^3;
 9*173+4 = 1561 = (-5)^3 + 7^3 + 7^3 + 10^3;
 9*174+4 = 1570 = (-5)^3 + 7^3 + (-14)^3 + 16^3;
 9*175+4 = 1579 = 16^3 + 37^3 + 37^3 + (-47)^3;
 9*176+4 = 1588 = 40^3 + 55^3 + 70^3 + (-83)^3;
 9*177+4 = 1597 = 4^3 + 28^3 + 82^3 + (-83)^3;
 9*178+4 = 1606 = (-20)^3 + 46^3 + 49^3 + (-59)^3;
 9*179+4 = 1615 = 10^3 + (-11)^3 + (-17)^3 + 19^3;
 9*180+4 = 1624 = 4^3 + (-5)^3 + (-8)^3 + 13^3;
 9*181+4 = 1633 = (-8)^3 + 16^3 + 25^3 + (-26)^3;
 9*182+4 = 1642 = (-2)^3 + 13^3 + 13^3 + (-14)^3;
 9*183+4 = 1651 = 1^3 + 13^3 + 13^3 + (-14)^3;
 9*184+4 = 1660 = (-11)^3 + (-14)^3 + (-17)^3 + 22^3;
 9*185+4 = 1669 = 10^3 + 10^3 + 10^3 + (-11)^3;
 9*186+4 = 1678 = (-2)^3 + 7^3 + 7^3 + 10^3;
 9*187+4 = 1687 = 1^3 + 7^3 + 7^3 + 10^3;
 9*188+4 = 1696 = 1^3 + 7^3 + (-14)^3 + 16^3;
 9*189+4 = 1705 = 16^3 + 16^3 + 46^3 + (-47)^3;
 9*190+4 = 1714 = 4^3 + 13^3 + 13^3 + (-14)^3;
 9*191+4 = 1723 = 7^3 + 13^3 + 16^3 + (-17)^3;
 9*192+4 = 1732 = (-11)^3 + (-11)^3 + 13^3 + 13^3;
 9*193+4 = 1741 = (-2)^3 + 4^3 + (-8)^3 + 13^3;
 9*194+4 = 1750 = 4^3 + 7^3 + 7^3 + 10^3;
 9*195+4 = 1759 = 4^3 + 7^3 + (-14)^3 + 16^3;
 9*196+4 = 1768 = 10^3 + 28^3 + 40^3 + (-44)^3;
 9*197+4 = 1777 = 7^3 + (-11)^3 + (-11)^3 + 16^3;
 9*198+4 = 1786 = 1^3 + (-20)^3 + (-23)^3 + 28^3;
 9*199+4 = 1795 = (-5)^3 + (-26)^3 + (-56)^3 + 58^3;
 9*200+4 = 1804 = (-23)^3 + (-32)^3 + (-32)^3 + 43^3;
 9*201+4 = 1813 = 4^3 + 4^3 + (-8)^3 + 13^3;
 9*202+4 = 1822 = (-5)^3 + (-5)^3 + (-5)^3 + 13^3;
 9*203+4 = 1831 = 7^3 + (-8)^3 + 10^3 + 10^3;
 9*204+4 = 1840 = (-8)^3 + 10^3 + (-14)^3 + 16^3;
 9*205+4 = 1849 = 4^3 + (-20)^3 + (-23)^3 + 28^3;
 9*206+4 = 1858 = (-2)^3 + 10^3 + (-11)^3 + 13^3;
 9*207+4 = 1867 = (-2)^3 + (-5)^3 + 10^3 + 10^3;
 9*208+4 = 1876 = 1^3 + (-5)^3 + 10^3 + 10^3;
 9*209+4 = 1885 = 4^3 + (-5)^3 + (-17)^3 + 19^3;
 9*210+4 = 1894 = 7^3 + 19^3 + 19^3 + (-23)^3;
 9*211+4 = 1903 = (-5)^3 + 7^3 + (-8)^3 + 13^3;
 9*212+4 = 1912 = (-2)^3 + (-26)^3 + (-56)^3 + 58^3;
 9*213+4 = 1921 = 1^3 + (-26)^3 + (-56)^3 + 58^3;
 9*214+4 = 1930 = 4^3 + 10^3 + (-11)^3 + 13^3;
 9*215+4 = 1939 = 4^3 + (-5)^3 + 10^3 + 10^3;
 9*216+4 = 1948 = 1^3 + (-5)^3 + (-5)^3 + 13^3;
 9*217+4 = 1957 = 13^3 + 13^3 + 28^3 + (-29)^3;
 9*218+4 = 1966 = 7^3 + (-17)^3 + (-32)^3 + 34^3;
 9*219+4 = 1975 = 7^3 + (-14)^3 + (-26)^3 + 28^3;
 9*220+4 = 1984 = (-2)^3 + (-2)^3 + 10^3 + 10^3;
 9*221+4 = 1993 = 1^3 + (-2)^3 + 10^3 + 10^3;
 9*222+4 = 2002 = 1^3 + 1^3 + 10^3 + 10^3;
 9*223+4 = 2011 = 4^3 + (-5)^3 + (-5)^3 + 13^3;
 9*224+4 = 2020 = (-2)^3 + 7^3 + (-8)^3 + 13^3;
 9*225+4 = 2029 = 7^3 + 7^3 + 7^3 + 10^3;
 9*226+4 = 2038 = 7^3 + 7^3 + (-14)^3 + 16^3;
 9*227+4 = 2047 = (-38)^3 + (-56)^3 + (-122)^3 + 127^3;
 9*228+4 = 2056 = (-2)^3 + 4^3 + 10^3 + 10^3;
 9*229+4 = 2065 = 1^3 + 4^3 + 10^3 + 10^3;
 9*230+4 = 2074 = 1^3 + 1^3 + (-5)^3 + 13^3;
 9*231+4 = 2083 = (-2)^3 + 55^3 + 115^3 + (-119)^3;
 9*232+4 = 2092 = 4^3 + 7^3 + (-8)^3 + 13^3;
 9*233+4 = 2101 = 13^3 + (-14)^3 + (-20)^3 + 22^3;
 9*234+4 = 2110 = (-11)^3 + (-23)^3 + (-50)^3 + 52^3;
 9*235+4 = 2119 = (-2)^3 + (-11)^3 + (-23)^3 + 25^3;
 9*236+4 = 2128 = 4^3 + 4^3 + 10^3 + 10^3;
 9*237+4 = 2137 = 1^3 + 4^3 + (-5)^3 + 13^3;
 9*238+4 = 2146 = 1^3 + 16^3 + 25^3 + (-26)^3;
 9*239+4 = 2155 = 7^3 + (-53)^3 + (-62)^3 + 73^3;
 9*240+4 = 2164 = (-5)^3 + 7^3 + (-17)^3 + 19^3;
 9*241+4 = 2173 = (-2)^3 + (-2)^3 + (-2)^3 + 13^3;
 9*242+4 = 2182 = 1^3 + (-2)^3 + (-2)^3 + 13^3;
 9*243+4 = 2191 = 1^3 + 1^3 + (-2)^3 + 13^3;
 9*244+4 = 2200 = 1^3 + 1^3 + 1^3 + 13^3;
 9*245+4 = 2209 = 7^3 + 10^3 + (-11)^3 + 13^3;
 9*246+4 = 2218 = (-5)^3 + 7^3 + 10^3 + 10^3;
 9*247+4 = 2227 = (-5)^3 + 10^3 + (-14)^3 + 16^3;
 9*248+4 = 2236 = (-8)^3 + (-23)^3 + (-29)^3 + 34^3;
 9*249+4 = 2245 = (-2)^3 + (-2)^3 + 4^3 + 13^3;
 9*250+4 = 2254 = 1^3 + (-2)^3 + 4^3 + 13^3;
 9*251+4 = 2263 = 1^3 + 1^3 + 4^3 + 13^3;
 9*252+4 = 2272 = (-8)^3 + (-11)^3 + (-14)^3 + 19^3;
 9*253+4 = 2281 = (-2)^3 + 7^3 + (-17)^3 + 19^3;
 9*254+4 = 2290 = (-5)^3 + (-5)^3 + 7^3 + 13^3;
 9*255+4 = 2299 = 7^3 + 25^3 + 67^3 + (-68)^3;
 9*256+4 = 2308 = 22^3 + 34^3 + 61^3 + (-65)^3;
 9*257+4 = 2317 = (-2)^3 + 4^3 + 4^3 + 13^3;
 9*258+4 = 2326 = 1^3 + 4^3 + 4^3 + 13^3;
 9*259+4 = 2335 = (-2)^3 + 7^3 + 10^3 + 10^3;
 9*260+4 = 2344 = 1^3 + 7^3 + 10^3 + 10^3;
 9*261+4 = 2353 = 1^3 + 10^3 + (-14)^3 + 16^3;
 9*262+4 = 2362 = (-11)^3 + 118^3 + 160^3 + (-179)^3;
 9*263+4 = 2371 = 7^3 + 7^3 + (-8)^3 + 13^3;
 9*264+4 = 2380 = 10^3 + 13^3 + 16^3 + (-17)^3;
 9*265+4 = 2389 = 4^3 + 4^3 + 4^3 + 13^3;
 9*266+4 = 2398 = (-5)^3 + (-5)^3 + (-20)^3 + 22^3;
 9*267+4 = 2407 = 4^3 + 7^3 + 10^3 + 10^3;
 9*268+4 = 2416 = 1^3 + (-5)^3 + 7^3 + 13^3;
 9*269+4 = 2425 = 25^3 + (-26)^3 + (-26)^3 + 28^3;
 9*270+4 = 2434 = 10^3 + (-11)^3 + (-11)^3 + 16^3;
 ...
$\endgroup$
  • $\begingroup$ So no numbers were skipped, right? $\endgroup$ – qwr Apr 24 '18 at 18:48
  • $\begingroup$ @qwr: yes, no numbers were skipped in the small range $k=1,2,...,280$ (in this case). And probably there is no general pattern for $4$-cubes formula to cover all the range $k\in\mathbb{N}$. $\endgroup$ – Oleg567 Apr 25 '18 at 3:19

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