Digital Sums
by Marek Ctrnact

I am Marek 14 from Czech Republic and I would like to tell you about some interesting things I found out about digital powers.

What is a digital power? Well, to get a nth digital power of an integer x, you have to count nth power of all digits in x and add it together!
Now, number 153 (for example) is a very interesting, because it is equal its third digital power: 1^3+5^3+3^3! Let's call such numbers "fixed digital cubes"  (or FD3s) When I first found about this, I tried to find some more numbers like this - there are 3 more: 370, 371 and 407.
What about 2nd power? Strangely enough, there is no fixed digital square (except from trivial solutions 0, 1). At least not in decimal system - in some other systems there are some (like 24 in octal system - 2^2+4^2=20=24(8))
Finally I made a program to search for numbers equivalent to set digital power in a set number system, but there are still some problems I would like to know answers, like if there is infinitely many n for which there is no fixed digital nth power other than 0, 1? It holds in binary system, but that is rather trivial result.

3 2
12(5) 22(8)

3 3
122(17)

3 5
1020(33) 1021(34) 2102(65)

3 6
2110(66) 2111(67) 11212(131)

3 7
112022(386) 200222(512)

3 8
100120(258) 100121(259)

3 9
201000(513) 201001(514) 2210220(2049) 2210221(2050)

3 10
11020000(1026) 1102001(1027) 11012220(3075) 11012221(3076) 12121212(4100)

3 12
211110222(16388)

3 13
2220122220(57345) 2220122221(57346)

4 3
20(8) 21(9) 130(28) 131(29) 203(35) 223(43) 313(55) 332(62)

4 4
1103(83) 3303(243)

4 5
200(32) 201(33)

4 6
32122(922)

4 7
2000(128) 2001(129) 210030(2316) 210031(2317) 212030(2444) 212031(2445) 220023(2571) 222023(2699)

4 8
1302202(7330) 3031010(13124) 3031011(13125)

4 9
20000(512) 20001(513) 11003212(20710) 11023212(21222) 21313020(40392) 21313021(40393)

4 11
200000(2048) 200001(2049)

4 12
10012133212(1075174)

5 2
23(13) 33(18)

5 3
103(28) 433(118)

5 4
2124(289) 2403(353) 3134(419)

5 5
2213(308)

5 6
124030(4890) 124031(4891) 242423(9113)

5 7
2012(257)

5 8
4112222(66562)

5 9
40302332(322217) 434434444(1874374)

6 3
243(99) 514(190) 1055(251)

6 5
1232(308) 14340(2292) 14341(2293) 14432(2324) 23520(3432) 23521(3433) 44405(6197)

6 6
435152(36140)

6 7
5435254(269458)

6 8
12222215(391907)

6 9
555435035(10067135)

7 2
13(10) 34(25) 44(32) 63(45)

7 3
12(9) 22(16) 250(133) 251(134) 305(152) 505(250)

7 5
122(65) 4332(1542) 12205(3190) 12252(3222) 13350(3612) 13351(3613) 15124(4183) 36034(9286)

7 6
205145(35411)

7 7
213443(37271) 1424553(191334) 1433554(193393) 3126542(376889) 4355653(535069) 6515652(794376)

7 8
422302(72865)

7 9
25003242(2236488) 34454124(3021897) 52445002(4431562) 125543055(8094840) 161340144(10883814) 254603255(16219922) 336133614(20496270) 542662326(32469576) 565264226(34403018)

7 10
12304122(1110699) 150012332(9885773) ?106571880

8 2
24(20) 64(52)

8 3
134(92) 205(133) 463(307) 660(432) 661(443)

8 4
20(16) 21(17) 400(256) 401(257) 420(272) 421(273)

8 5
2040(1056) 2041(1057) 4423(2323) 40663(16819) 42710(17864) 42711(17865) 60007(24583) 62047(25639)

8 6
636703(212419)

8 7
200(128) 201(129) 40000(16384) 40001(16385) 40200(16512) 40201(16513) 3352072(906298) 3352272(906426) 3451473(938811) 4217603(1122179) 7755336(2087646)

8 8
32103(13379) 200400(65792) 200401(65793) 16450603(3821955) 63717005(13606405)

8 9
233173324(40695508)

9 2
45(41) 55(50)

9 3
30(27) 31(28) 150(126) 151(127) 570(468) 571(469) 1388(1052)

9 4
432(353) 2446(1824)

9 5
300(243) 301(244) 12036(8052) 12336(8295) 14462(9857) 108810(65538) 108811(65539)

9 6
55433(36804)

9 7
3000(2187) 3001(2188) 2225764(1198372) 6275850(3357009) 6275851(3357010) 10867328(5300099)

9 8
12742452(6287267)

10 3
153 370 371 407

10 4
1634 8208 9474

10 5
4150 4151 54748 92727 93084 194979

10 6
548834

10 7
1741725 4210818 9800817 9926315 14459929

11 2
05 06 (61) 06 06 (72)

11 3
03 02 (35) 01 00 05 (126) 03 00 07 (370) 07 00 08 (855) 10 00 06 (1216) 10 06 04 (1280)

11 4
02 04 02 (288) 08 00 00 09 (10657)

11 5
02 03 00 (275) 02 03 01 (276) 02 09 01 08 (3770) 03 05 03 04 (4635) 05 05 01 04 (7275) 05 05 04 02 (7306) 06 00 03 00 (8019) 01 01 07 02 00 (16841) 01 01 07 02 01 (16842) 01 02 04 07 00 (17864) 01 02 04 07 01 (17865) 02 05 08 04 00 (36949) 02 05 08 04 01 (36950) 04 03 09 03 05 (63684) 04 05 09 01 05 (66324) 04 09 05 06 03 (71217) 06 01 07 08 08 (90120) 06 08 09 01 00 (99594) 06 08 09 01 01 (99595) 09 07 02 08 08 (141424) 10 08 02 07 06 (157383) 01 02 06 10 08 09 (199626) 01 08 10 02 10 09 (291850)

11 6
07 02 06 03 10 00 (1165098) 07 02 06 03 10 01 (1165099)

12 2
02 05 (29) 10 05 (125)

12 3
05 07 07 (811) 06 06 08 (944) 10 08 03 (1539) 01 01 10 10 (2002)

12 5
01 04 07 06 05 (28733) 09 03 08 10 04 (193084)

13 2
01 04 (17) 03 06 (45) 06 07 (85) 07 07 (98) 10 06 (136) 12 04 (160)

13 3
04 09 00 (793) 04 09 01 (794) 05 00 09 (854) 11 08 05 (1968)

13 4
03 09 06 04 (8194)

14 3
01 03 06 (244) 04 00 09 (793)

15 2
07 08 (113) 08 08 (128)