World!Of
Numbers

WON plate
187 |

[ January 2, 2014 ] [ Last update November 23, 2015 ]
Expressing nine- or pandigitals as sums of powers of 2 and/or 3
in as few terms as possible

Take care exponents greater than or equal to 2.

Inspired by B.S. Rangaswamy's work allow me to present here
a personal WONplate. I will compile here the best 100 or so contributions
with minimum record solutions. In total there are 9! or 362880 ninedigitals &
9.9! or 3265920 pandigitals, enough to choose from. To set things in motion
I will start with the first and last one and a few in-betweens
that are already contributions from B.S. Rangaswamy, the master himself!
Note : repetitions of one or more powers are allowed but
each one is included in the final 'tier' count.

challenge 1 find all ninedigitals with only three or four terms.
challenge 2 find all pandigitals with only three or four terms.

On october 7, 2015 I (PDG) wrote a ubasic program
and here are already some statistics:

No ninedigital can be expressed with only two terms.
There is exactly one unique ninedigital that can be expressed with three terms.
There are exactly 30 ninedigitals that can be expressed with four terms.

No pandigital can be expressed with only two terms.
There are exactly 5 pandigitals that can be expressed with three terms.
There are exactly 48 pandigitals that can be expressed with four terms.

Ninedigitals
129378546 317 + 311 + 310 + 374_tierPDG
129736548 317 + 219 + 216 + 384_tierPDG
134258697 227 + 215 + 213 + 324_tierPDG
134258769 227 + 215 + 213 + 344_tierPDG
134258796 227 + 215 + 213 + 34 + 335_tierBSR
134258976 227 + 215 + 213 + 28 + 255_tierBSR
134265798 227 + ( 39 + 39 ) + 213 + 295_tierBSR
134279586 227 + 310 + 211 + 36 + 255_tierBSR
134285967 227 + 216 + 37 + 29 + 225_tierBSR
134286579 227 + 216 + 211 + 210 + 355_tierBSR
134287569 227 + 216 + 212 + 27 + 345_tierBSR
134289675 227 + 216 + 212 + 37 + 275_tierBSR
134759826 227 + 312 + 38 + 2124_tierPDG
136845729 227 + 221 + 219 + 384_tierPDG
143562798 317 + 315 + 216 + 2134_tierPDG
268435719 228 + 35 + 24 + 224_tierBSR
268435971 228 + 28 + 35 + 244_tierBSR
268437915 228 + 37 + 28 + 244_tierPDG
268451937 228 + 214 + 34 + 244_tierPDG
268471539 228 + 39 + 214 + 244_tierPDG
269485137 228 + 220 + 210 + 344_tierPDG
387421956 318 + ( 36 + 36 ) + 324_tierPDG
387952416 318 + 312 + ( 35 + 35 )4_tierPDG
387954126 318 + 312 + 37 + 324_tierPDG
389521746 318 + 221 + 212 + 324_tierPDG
389521764 318 + 221 + 212 + 334_tierPDG
392158764 318 + 222 + 219 + 394_tierPDG
536871249 229 + 28 + 34                     (unique!)3_tierBSR
536871492 229 + 29 + 26 + 22
OR
229 + 28 + 35 + 34
4_tierBSR

PDG
536871924 229 + 36 + 28 + 334_tierBSR
536874219 229 + 37 + 210 + 26 + 255_tierBSR
536879124 229 + 213 + 24 + 224_tierBSR
536879241 229 + 213 + 27 + 324_tierBSR
536897412 229 + 39 + 38 + 284_tierPDG
537198624 229 + 218 + 216 + 254_tierPDG
537921684 229 + 220 + 37 + 324_tierPDG
538467291 229 + 313 + 211 + 234_tierPDG
538972164 229 + 221 + 212 + 224_tierPDG
538972416 229 + 221 + 212 + 284_tierPDG
647295138 318 + ( 317 + 317 ) + 3134_tierPDG

Pandigitals
1073952468 230 + 311 + 215 + 364_tierBSR
1074289653 230 + 312 + 214 + 224_tierPDG
1074659328 230 + 219 + 218 + 2174_tierPDG
1075368924 230 + 313 + 215 + 324_tierPDG
1075368942 230 + 313 + 215 + 334_tierPDG
1075839246 230 + 221 + 35 + 334_tierPDG
1075839264 230 + 221 + 28 + 254_tierPDG
1075839462 230 + 221 + ( 35 + 35 )4_tierPDG
1075986432 230 + 221 + 217 + 2144_tierPDG
1208549376 230 + 227 + 219 + 2164_tierPDG
1248357096 319 + ( 316 + 316 ) + 374_tierPDG
1305927684 319 + 317 + 315 + 3114_tierPDG
1324956078 319 + 317 + 225 + 244_tierPDG
1324957086 319 + 317 + 225 + 2104_tierPDG
1342970865 230 + 228 + 312 + 2184_tierPDG
1463259708 230 + 318 + 221 + 354_tierPDG
1549682037 319 + 318 + 34                              3_tierBSR
1549702368 319 + 318 + 39 + 364_tierPDG
1549703826 319 + 318 + 39 + 374_tierPDG
1587430692 319 + 318 + 225 + 2224_tierPDG
1592730864 319 + 318 + 316 + 374_tierPDG
1592748360 319 + 318 + 316 + 394_tierPDG
2148035796 231 + 312 + 39 + 2104_tierPDG
2319670548 231 + 317 + 316 + 244_tierPDG
2534971860 231 + 318 + 216 + 374_tierPDG
2578016394 231 + 318 + 316 + 2164_tierPDG
3486791052 320 + 38 + 34 + 324_tierBSR
3486791205 320 + 38 + 353_tierBSR
3486791250 320 + 38 + 28 + 254_tierBSR
3486915720 320 + 217 + 35 + 224_tierPDG
3489012657 320 + 221 + 217 + 254_tierPDG
3491570286 320 + 314 + 37 + 364_tierPDG
3491587062 320 + 314 + 39 + 324_tierPDG
3492157860 320 + 314 + 312 + 3104_tierPDG
3495761802 320 + 314 + 222 + 274_tierPDG
3615924708 320 + 317 + 27 + 244_tierPDG
3615924807 320 + 317 + 353_tierPDG
3615927480 320 + 317 + 37 + 364_tierPDG
3658972014 320 + 317 + 316 + 364_tierPDG
3874205619 320 + 318 + 363_tierBSR
4295018739 232 + 215 + 214 + 211 + 355_tierBSR
4295019783 232 + 215 + 39 + 25 + 225_tierBSR
4295163078 232 + 311 + 214 + 37 + 265_tierBSR
4329570816 232 + 225 + 220 + 294_tierPDG
5036472918 320 + 319 + 318 + 384_tierPDG
5368709124 232 + 230 + 223_tierBSR
5368741920 232 + 230 + 215 + 254_tierPDG
5369240817 232 + 230 + 312 + 284_tierPDG
6978352014 ( 320 + 320 ) + 314 + 354_tierPDG
8590136247 233 + 311 + 39 + 212 + 365_tierBSR
8590216437 233 + 218 + 39 + 32 + 325_tierBSR
8590216473 233 + 218 + 39 + 33 + 335_tierBSR
8590374126 233 + 218 + 311 + 354_tierBSR
8590374612 233 + 218 + 311 + 364_tierBSR
8590374621 233 + 218 + 311 + 36 + 325_tierBSR
8590461327 233 + 219 + 37 + 28 + 225_tierBSR
8590463721 233 + 219 + 212 + 36 + 245_tierBSR
8590472316 233 + 312 + 212 + 374_tierBSR
8591247360 233 + 220 + 218 + 2114_tierPDG
8592031764 233 + 221 + 24 + 224_tierPDG
8724160539 233 + 227 + 213 + 334_tierPDG
8724160593 233 + 227 + 213 + 344_tierPDG

Ubasic code to find all the 4_tiers solutions
```
10   dim P(51),Q(51)
20   P(1)=2^2:Q(1)="2^2"
30   P(2)=2^3:Q(2)="2^3"
40   P(3)=2^4:Q(3)="2^4"
50   P(4)=2^5:Q(4)="2^5"
60   P(5)=2^6:Q(5)="2^6"
70   P(6)=2^7:Q(6)="2^7"
80   P(7)=2^8:Q(7)="2^8"
90   P(8)=2^9:Q(8)="2^9"
100   P(9)=2^10:Q(9)="2^10"
110   P(10)=2^11:Q(10)="2^11"
120   P(11)=2^12:Q(11)="2^12"
130   P(12)=2^13:Q(12)="2^13"
140   P(13)=2^14:Q(13)="2^14"
150   P(14)=2^15:Q(14)="2^15"
160   P(15)=2^16:Q(15)="2^16"
170   P(16)=2^17:Q(16)="2^17"
180   P(17)=2^18:Q(17)="2^18"
190   P(18)=2^19:Q(18)="2^19"
200   P(19)=2^20:Q(19)="2^20"
210   P(20)=2^21:Q(20)="2^21"
220   P(21)=2^22:Q(21)="2^22"
230   P(22)=2^23:Q(22)="2^23"
240   P(23)=2^24:Q(23)="2^24"
250   P(24)=2^25:Q(24)="2^25"
260   P(25)=2^26:Q(25)="2^26"
270   P(26)=2^27:Q(26)="2^27"
280   P(27)=2^28:Q(27)="2^28"
290   P(28)=2^29:Q(28)="2^29"
300   P(29)=2^30:Q(29)="2^30"
310   P(30)=2^31:Q(30)="2^31"
320   P(31)=2^32:Q(31)="2^32"
330   P(32)=2^33:Q(32)="2^33"
340   P(33)=3^2:Q(33)="3^2"
350   P(34)=3^3:Q(34)="3^3"
360   P(35)=3^4:Q(35)="3^4"
370   P(36)=3^5:Q(36)="3^5"
380   P(37)=3^6:Q(37)="3^6"
390   P(38)=3^7:Q(38)="3^7"
400   P(39)=3^8:Q(39)="3^8"
410   P(40)=3^9:Q(40)="3^9"
420   P(41)=3^10:Q(41)="3^10"
430   P(42)=3^11:Q(42)="3^11"
440   P(43)=3^12:Q(43)="3^12"
450   P(44)=3^13:Q(44)="3^13"
460   P(45)=3^14:Q(45)="3^14"
470   P(46)=3^15:Q(46)="3^15"
480   P(47)=3^16:Q(47)="3^16"
490   P(48)=3^17:Q(48)="3^17"
500   P(49)=3^18:Q(49)="3^18"
510   P(50)=3^19:Q(50)="3^19"
520   P(51)=3^20:Q(51)="3^20"
530   open "out187-4.txt" for append as #1
540   for Loopa=1 to 51
550   Na=P(Loopa)
560   for Loopb=1 to 51
570   Nb=P(Loopb)
580   for Loopc=1 to 51
590   Nc=P(Loopc)
600   for Loopd=1 to 51
610   Nd=P(Loopd):Nin=0:Pan=0
620   Kand=cutspc(str(Na+Nb+Nc+Nd))
630   '
640   Lenkand=len(Kand)
650   if Lenkand<>9 then goto 690
660   Nin=1:for I=1 to 9:Nin*=prm(val(mid(Kand,I))):next I
670   if Nin=223092870 then print Kand;" = ";Q(Loopa);" + ";Q(Loopb);" + ";Q(L
oopc);" + ";Q(Loopd):print #1,Kand;" = ";Q(Loopa);" + ";Q(Loopb);" + ";Q(Loopc);
" + ";Q(Loopd)
680   goto 730
690   if Lenkand<>10 then goto 730
700   Pan=1:for I=1 to 10:Pan*=prm(val(mid(Kand,I))):next I
710   if Pan=223092870 then print Kand;" = ";Q(Loopa);" + ";Q(Loopb);" + ";Q(L
oopc);" + ";Q(Loopd):print #1,Kand;" = ";Q(Loopa);" + ";Q(Loopb);" + ";Q(Loopc);
" + ";Q(Loopd)
720   '
730   print Loopa;Loopb;Loopc;Loopd,Kand,Nin;Pan
740   next Loopd
750   next Loopc
760   next Loopb
770   next Loopa

```

Some of the following solutions with tiers greater than 6
could be promoted i.e. could be expressed with fewer terms.
I don't claim to have displayed the solutions with least terms.

Some Ninedigitals with tiers greater than 5
123456789 226 + 316 + 223 + 314 + 216 + 310 + 212 + 36 +
27 + 34 + 23
11_tierPDG
134256789 227 + 215 + 212 + 211 + 34 + 26 + 227_tierPDG
987654321 229 + 318 + 316 + 224 + 221 + 220 + 218 + 217 +
33 + 23 + 22
11_tierPDG

Some Pandigitals with tiers greater than 5
1023456789 229 + 318 + 226 + 224 + 315 + 312 + 218 + 217 + 212 + 210 + 29 +
34 + 33 + 22
14_tierBSR
1073824569 230 + 216 + 214 + 36 + 26 + 256_tierBSR
1073824695 230 + 310 + 39 + 212 + 33 + 246_tierBSR
1073892654 230 + 217 + 39 + 25 + 33 + 246_tierBSR
1073894652 230 + 217 + 39 + 211 + 24 + 326_tierBSR
1073925684 230 + 311 + 38 + 27 + 24 + 236_tierBSR
1073925864 230 + 311 + 38 + 35 + 34 + 236_tierBSR
1073952486 230 + 311 + 215 + 36 + 32 + 326_tierBSR
1073954286 230 + 311 + 215 + 211 + 28 + 356_tierBSR
1073958462 230 + 311 + 215 + 38 + 34 + 346_tierBSR
1073984652 230 + 311 + 216 + 27 + 32 + 236_tierBSR
3486795201 330 + 213 + 211 + 29 + 25 + 246_tierBSR
4295137806 232 + 217 + 39 + 39 + 26 + 236_tierBSR
4295138706 232 + 217 + 39 + 39 + 36 + 356_tierBSR
4295163807 232 + 311 + 214 + 37 + 36 + 266_tierBSR
8590126743 233 + 311 + 213 + 38 + 35 + 236_tierBSR
8590213467 233 + 218 + 214 + 28 + 26 + 336_tierBSR
8590216734 233 + 218 + 39 + 35 + 26 + 236_tierBSR
8590216743 233 + 218 + 39 + 28 + 26 + 226_tierBSR
8590217436 233 + 218 + 39 + 36 + 28 + 256_tierBSR
8590217463 233 + 218 + 39 + 210 + 24 + 226_tierBSR
8590217634 233 + 218 + 39 + 36 + 35 + 356_tierBSR
8590231764 233 + 218 + 215 + 37 + 26 + 326_tierBSR
8590234176 233 + 218 + 215 + 212 + 29 + 266_tierBSR
8590236174 233 + 218 + 39 + 39 + 26 + 236_tierBSR
8590264731 233 + 218 + 216 + 37 + 28 + 246_tierBSR
8590271436 233 + 218 + 216 + 213 + 36 + 356_tierBSR
8590463172 233 + 219 + 212 + 27 + 26 + 226_tierBSR
8590471236 233 + 219 + 213 + 212 + 26 + 226_tierBSR
8590476312 233 + 219 + 214 + 210 + 24 + 236_tierBSR
9876543210 233 + 230 + 227 + 226 + 223 + 221 + 220 + 212 + 210 +
36 + 32 + 23
12_tierBSR

Inspired by Mr Albert H Beiler, who discovered the following 30 ninedigital squares
and 87 pandigital squares (see left column in the resp. tables), B.S. Rangaswamy
shows their respective constituent powers of 2 and/or 3 after manual manipulations.

30 Ninedigital Squares as sum of powers of 2 and/or 3 (all by BSR)
(11826)2 =
139854276
[ 1 ] 10_tier
227 + 314 + 312 + 218 + 310 + 36 + 27 + 26 + 24 + 23
(12363)2 =
152843769
[ 2 ] 11_tier
227 + 224 + 313 + 311 + 216 + 213 + 37 + 210 + 28 + 27 + 25
(12543)2 =
157326849
[ 3 ] 12_tier
227 + 224 + 314 + 220 + 218 + 311 + 310 + 210 + 36 + 35 + 24 + 23
(14676)2 =
215384976
[ 4 ] 13_tier
227 + 226 + 223 + 314 + 312 + 218 + 216 + 39 + 38 + 210 +
28 + 34 + 34
(15681)2 =
245893761
[ 5 ] 12_tier
227 + 226 + 316 + 220 + 218 + 311 + 39 + 38 + 212 + 37 + 33 + 33
(15963)2 =
254817369
[ 6 ] 13_tier
227 + 226 + 316 + 223 + 313 + 218 + 311 + 39 + 211 + 26 +
33 + 23 + 22
(18072)2 =
326597184
[ 7 ] 12_tier
228 + 316 + 315 + 312 + 311 + 215 + 39 + 37 + 211 + 36 + 34 + 24
(19023)2 =
361874529
[ 8 ] 11_tier
228 + 226 + 224 + 223 + 220 + 216 + 215 + 214 + 210 + 34 + 24
(19377)2 =
375468129
[ 9 ] 12_tier
228 + 226 + 225 + 314 + 220 + 219 + 38 + 38 + 28 + 34 + 34 + 22
(19569)2 =
382945761
[ 10 ] 12_tier
228 + 226 + 316 + 222 + 217 + 39 + 38 + 211 + 36 + 35 + 26 + 24
(19629)2 =
385297641
[ 11 ] 12_tier
228 + 226 + 316 + 314 + 313 + 218 + 216 + 36 + 29 + 35 + 27 + 24
(20316)2 =
412739856
[ 12 ] 9_tier
318 + 224 + 223 + 217 + 39 + 37 + 29 + 34 + 23
(22887)2 =
523814769
[ 13 ] 9_tier
318 + 227 + 221 + 216 + 38 + 38 + 36 + 32 + 22
(23019)2 =
529874361
[ 14 ] 11_tier
318 + 227 + 314 + 221 + 220 + 218 + 215 + 213 + 212 + 35 + 22
(23178)2 =
537219684
[ 15 ] 9_tier
229 + 218 + 216 + 39 + 36 + 29 + 27 + 25 + 23
(23439)2 =
549386721
[ 16 ] 12_tier
229 + 223 + 221 + 313 + 218 + 217 + 215 + 38 + 37 + 36 + 28 + 32
(24237)2 =
587432169
[ 17 ] 12_tier
229 + 316 + 314 + 221 + 312 + 216 + 215 + 37 + 37 + 28 + 25 + 23
(24276)2 =
589324176
[ 18 ] 13_tier
229 + 316 + 223 + 312 + 218 + 311 + 215 + 213 + 212 + 211 +
34 + 32 + 32
(24441)2 =
597362481
[ 19 ] 9_tier
229 + 316 + 224 + 312 + 217 + 37 + 37 + 36 + 24
(24807)2 =
615387249
[ 20 ] 10_tier
229 + 226 + 223 + 221 + 219 + 218 + 217 + 212 + 34 + 25
(25059)2 =
627953481
[ 21 ] 12_tier
229 + 226 + 224 + 314 + 221 + 218 + 215 + 39 + 210 + 36 + 24 + 22
(25572)2 =
653927184
[ 22 ] 9_tier
229 + 226 + 316 + 314 + 221 + 39 + 29 + 35 + 27
(25941)2 =
672935481
[ 23 ] 10_tier
229 + 227 + 313 + 217 + 310 + 310 + 37 + 210 + 27 + 32
(26409)2 =
697435281
[ 24 ] 10_tier
229 + 227 + 224 + 223 + 220 + 217 + 210 + 27 + 32 + 23
(26733)2 =
714653289
[ 25 ] 11_tier
229 + 227 + 316 + 218 + 311 + 310 + 214 + 211 + 210 + 27 + 22
(27129)2 =
735982641
[ 26 ] 12_tier
229 + 317 + 226 + 221 + 312 + 311 + 215 + 39 + 37 + 37 + 27 + 32
(27273)2 =
743816529
[ 27 ] 12_tier
229 + 227 + 226 + 314 + 312 + 218 + 215 + 213 + 36 + 29 + 35 + 33
(29034)2 =
842973156
[ 28 ] 13_tier
229 + 228 + 225 + 221 + 313 + 218 + 217 + 214 + 213 + 211 +
210 + 32 + 23
(29106)2 =
847159236
[ 29 ] 14_tier
229 + 228 + 225 + 314 + 221 + 220 + 218 + 216 + 215 + 38 + 37 +
29 + 33 + 22
(30384)2 =
923187456
[ 30 ] 14_tier
229 + 228 + 226 + 316 + 314 + 221 + 312 + 218 + 215 + 214 + 211 +
29 + 34 + 22

87 Pandigital Squares as sum of powers of 2 and/or 3 (all by BSR)
(32043)2 =
1026753849
[ 1 ] 11_tier
229 + 318 + 226 + 225 + 313 + 217 + 216 + 213 + 24 + 32 + 22
(32286)2 =
1042385796
[ 2 ] 13_tier
229 + 318 + 226 + 316 + 314 + 221 + 220 + 38 + 37 + 210 +
28 + 34 + 22
(33144)2 =
1098524736
[ 3 ] 12_tier
230 + 224 + 314 + 221 + 220 + 216 + 38 + 212 + 36 + 26 + 32 + 22
(35172)2 =
1237069584
[ 4 ] 13_tier
319 + 226 + 314 + 221 + 312 + 218 + 39 + 212 + 36 + 36 +
28 + 33 + 33
(35337)2 =
1248703569
[ 5 ] 10_tier
319 + 226 + 224 + 221 + 218 + 217 + 216 + 26 + 33 + 33
(35757)2 =
1278563049
[ 6 ] 13_tier
319 + 226 + 316 + 314 + 220 + 218 + 215 + 214 + 211 + 210 +
26 + 24 + 22
(35853)2 =
1285437609
[ 7 ] 13_tier
319 + 226 + 316 + 223 + 222 + 218 + 217 + 215 + 38 + 212 +
36 + 35 + 25
(37176)2 =
1382054976
[ 8 ] 13_tier
319 + 227 + 226 + 224 + 313 + 216 + 39 + 38 + 37 + 210 +
35 + 27 + 24
(37905)2 =
1436789025
[ 9 ] 10_tier
319 + 228 + 314 + 220 + 311 + 216 + 214 + 36 + 36 + 25
(38772)2 =
1503267984
[ 10 ] 11_tier
319 + 228 + 226 + 314 + 312 + 217 + 214 + 35 + 26 + 24 + 23
(39147)2 =
1532487609
[ 11 ] 10_tier
319 + 228 + 226 + 225 + 220 + 310 + 39 + 26 + 32 + 32
(39336)2 =
1547320896
[ 12 ] 13_tier
319 + 228 + 226 + 316 + 314 + 313 + 310 + 39 + 213 + 212 +
26 + 23 + 22
(40545)2 =
1643897025
[ 13 ] 14_tier
319 + 318 + 226 + 224 + 223 + 313 + 218 + 216 + 214 + 36 + 36 +
29 + 24 + 23
(42744)2 =
1827049536
[ 14 ] 13_tier
319 + 229 + 226 + 316 + 224 + 312 + 218 + 311 + 38 + 38 +
35 + 35 + 24
(43902)2 =
1927385604
[ 15 ] 13_tier
230 + 229 + 228 + 316 + 314 + 218 + 311 + 216 + 37 + 29 +
27 + 26 + 22
(44016)2 =
1937408256
[ 16 ] 14_tier
230 + 229 + 228 + 316 + 315 + 312 + 218 + 217 + 215 + 38 + 28 +
34 + 34 + 25
(45567)2 =
2076351489
[ 17 ] 13_tier
319 + 229 + 228 + 226 + 225 + 314 + 221 + 220 + 217 + 310 +
210 + 29 + 22
(45624)2 =
2081549376
[ 18 ] 12_tier
319 + 229 + 228 + 226 + 316 + 221 + 313 + 217 + 211 +
210 + 28 + 34
(46587)2 =
2170348569
[ 19 ] 11_tier
319 + 229 + 318 + 226 + 315 + 221 + 311 + 310 + 212 + 35 + 35
(48852)2 =
2386517904
[ 20 ] 13_tier
231 + 227 + 226 + 225 + 221 + 313 + 218 + 217 + 216 + 37 +
36 + 34 + 23
(49314)2 =
2431870596
[ 21 ] 8_tier
231 + 228 + 315 + 313 + 213 + 33 + 33 + 24
(49353)2 =
2435718609
[ 22 ] 12_tier
231 + 228 + 224 + 313 + 220 + 218 + 216 + 215 + 214 +
37 + 35 + 27
(50706)2 =
2571098436
[ 23 ] 12_tier
319 + 319 + 227+ 226 + 316 + 221 + 216 + 215 + 38 +
34 + 26 + 33
(53976)2 =
2913408576
[ 24 ] 14_tier
231 + 229 + 227+ 226 + 224 + 223 + 221 + 218 + 311 + 214 +
38 + 37 + 24 + 32
(54918)2 =
3015986724
[ 25 ] 13_tier - Ending with Highest constituent power 29
231 + 229 + 228 + 316 + 224 + 221 + 220 + 217 + 216 + 39 +
213 + 211 + 29
(55446)2 =
3074258916
[ 26 ] 12_tier
231 + 229 + 318 + 221 + 218 + 310 + 310 + 212 + 211 +
28 + 26 + 32
(55524)2 =
3082914576
[ 27 ] 11_tier
231 + 229 + 318 + 223 + 221 + 312 + 216 + 215 + 39 + 212 + 35
(55581)2 =
3089247561
[ 28 ] 11_tier
231 + 229 + 318 + 224 + 312 + 217 + 39 + 38 + 212 + 37 + 28
(55626)2 =
3094251876
[ 29 ] 14_tier
231 + 229 + 318 + 224 + 314 + 312 + 218 + 310 + 310 + 37 + 37 +
29 + 26 + 32
(56532)2 =
3195867024
[ 30 ] 13_tier - Ending with Highest constituent power 29
231 + 229 + 318 + 226 + 316 + 223 + 314 + 312 + 311 + 215 +
214 + 38 + 29
(57321)2 =
3285697041
[ 31 ] 12_tier
231 + 230 + 316 + 224 + 222 + 218 + 217 + 310 + 210 +
33 + 23 + 22
(58413)2 =
3412078569
[ 32 ] 10_tier
231 + 319 + 226 + 225 + 313 + 216 + 213 + 211 + 25 + 33
(58455)2 =
3416987025
[ 33 ] 13_tier
231 + 319 + 226 + 225 + 314 + 313 + 311 + 39 + 212 + 28 +
27 + 23 + 22
(58554)2 =
3428570916
[ 34 ] 10_tier
231 + 319 + 226 + 316 + 223 + 218 + 214 + 211 + 210 + 23
(59403)2 =
3528716409
[ 35 ] 13_tier
231 + 319 + 227 + 226 + 224 + 219 + 218 + 216 + 213 + 38 +
36 + 25 + 22
(60984)2 =
3719048256
[ 36 ] 14_tier
320 + 227 + 226 + 224 + 223 + 314 + 312 + 218 + 311 + 214 + 36 +
29 + 34 + 25
(61575)2 =
3791480625
[ 37 ] 12_tier
320 + 228 + 225 + 221 + 219 + 310 + 39 + 212 + 211 + 24 + 22
(61866)2 =
3827401956
[ 38 ] 9_tier
320 + 228 + 226 + 314 + 218 + 39 + 213 + 35 + 22
(62679)2 =
3928657041
[ 39 ] 13_tier
320 + 318 + 316 + 223 + 221 + 312 + 218 + 216 + 310 + 36 +
29 + 35 + 24
(62961)2 =
3964087521
[ 40 ] 13_tier
320 + 318 + 226 + 224 + 314 + 220 + 217 + 215 + 36 + 35 +
34 + 34 + 25
(63051)2 =
3975428601
[ 41 ] 11_tier
320 + 318 + 226 + 225 + 312 + 39 + 38 + 37 + 29 + 33 + 22
(63129)2 =
3985270641
[ 42 ] 12_tier
320 + 318 + 226 + 316 + 312 + 218 + 310 + 215 + 39 +
212 + 36 + 28
(65634)2 =
4307821956
[ 43 ] 9_tier - All constituents are powers of 2 only.
232 + 223 + 222 + 218 + 213 + 210 + 28 + 27 + 22
(65637)2 =
4308215769
[ 44 ] 9_tier
232 + 223 + 314 + 216 + 213 + 211 + 210 + 26 + 25
(66105)2 =
4369871025
[ 45 ] 12_tier
232 + 226 + 314 + 221 + 312 + 218 + 310 + 310 + 37 +
36 + 34 + 26
(66276)2 =
4392508176
[ 46 ] 13_tier
232 + 226 + 224 + 223 + 314 + 218 + 311 + 215 + 38 + 212 +
28 + 35 + 23
(67677)2 =
4580176329
[ 47 ] 12_tier
232 + 228 + 315 + 221 + 218 + 310 + 212 + 211 + 34 +
26 + 25 + 22
(68763)2 =
4728350169
[ 48 ] 12_tier
232 + 318 + 316 + 221 + 312 + 218 + 39 + 212 + 36 + 28 + 34 + 34
(68781)2 =
4730825961
[ 49 ] 12_tier
232 + 318 + 316 + 314 + 312 + 310 + 214 + 210 + 29 +
26 + 23 + 22
(69513)2 =
4832057169
[ 50 ] 10_tier
232 + 229 + 311 + 215 + 213 + 29 + 35 + 34 + 32 + 32
(71433)2 =
5102673489
[ 51 ] 11_tier
232 + 229 + 228 + 221 + 218 + 215 + 38 + 210 + 27 + 25 + 24
(72621)2 =
5273809641
[ 52 ] 12_tier
320 + 230 + 229 + 227 + 225 + 223 + 311 + 216 + 213 +
36 + 27 + 22
(75759)2 =
5739426081
[ 53 ] 13_tier
232 + 319 + 228 + 223 + 314 + 312 + 215 + 214 + 38 + 211 +
210 + 25 + 33
(76047)2 =
5783146209
[ 54 ] 11_tier
232 + 319 + 228 + 316 + 315 + 216 + 214 + 212 + 28 + 34 + 32
(76182)2 =
5803697124
[ 55 ] 12_tier
232 + 319 + 228 + 226 + 223 + 221 + 218 + 217 + 215 +
213 + 212 + 32
(77346)2 =
5982403716
[ 56 ] 14_tier
232 + 319 + 318 + 227 + 221 + 220 + 218 + 216 + 310 + 37 + 211 +
33 + 32 + 23
(78072)2 =
6095237184
[ 57 ] 12_tier
232 + 319 + 229 + 226 + 225 + 218 + 311 + 215 + 211 +
34 + 24 + 32
(78453)2 =
6154873209
[ 58 ] 13_tier
232 + 319 + 229 + 227 + 224 + 223 + 220 + 218 + 310 + 39 +
29 + 32 + 32
(80361)2 =
6457890321
[ 59 ] 10_tier
232 + 231 + 315 + 220 + 39 + 39 + 37 + 28 + 34 + 22
(80445)2 =
6471398025
[ 60 ] 13_tier
232 + 231 + 224 + 223 + 221 + 313 + 216 + 39 + 212 + 35 +
27 + 26 + 25
(81222)2 =
6597013284
[ 61 ] 13_tier
232 + 231 + 227 + 224 + 221 + 220 + 218 + 217 + 39 + 38 +
211 + 27 + 25
(81945)2 =
6714983025
[ 62 ] 13_tier
232 + 231 + 228 + 221 + 313 + 218 + 217 + 213 + 37 + 36 +
36 + 34 + 24
(83919)2 =
7042398561
[ 63 ] 11_tier
320 + 320 + 226 + 313 + 216 + 310 + 36 + 36 + 29 + 32 + 23
(84648)2 =
7165283904
[ 64 ] 14_tier
320 + 320 + 227 + 316 + 223 + 314 + 220 + 311 + 215 + 214 +
212 + 34 + 24 + 23
(85353)2 =
7285134609
[ 65 ] 10_tier
320 + 320 + 228 + 316 + 310 + 39 + 212 + 36 + 26 + 32
(85743)2 =
7351862049
[ 66 ] 13_tier
232 + 231 + 229 + 228 + 226 + 225 + 221 + 220 + 218 + 215 +
36 + 26 + 23
(85803)2 =
7362154809
[ 67 ] 11_tier
320 + 320 + 318 + 220 + 216 + 215 + 214 + 37 + 25 + 33 + 23
(86073)2 =
7408561329
[ 68 ] 12_tier
320 + 320 + 318 + 316 + 222 + 218 + 216 + 37 + 210 +
34 + 25 + 32
(87639)2 =
7680594321
[ 69 ] 13_tier
320 + 320 + 229 + 227 + 225 + 221 + 218 + 39 + 37 + 36 +
29 + 25 + 23
(88623)2 =
7854036129
[ 70 ] 13_tier
232 + 320 + 226 + 314 + 218 + 216 + 310 + 212 + 210 + 36 +
32 + 23 + 22
(89079)2 =
7935068241
[ 71 ] 13_tier
232 + 320 + 227 + 224 + 221 + 311 + 215 + 38 + 38 + 210 +
35 + 27 + 24
(89145)2 =
7946831025
[ 72 ] 14_tier
232 + 320 + 227 + 224 + 223 + 314 + 312 + 218 + 310 + 310 +
210 + 26 + 25 + 22
(89355)2 =
7984316025
[ 73 ] 12_tier
232 + 320 + 227 + 226 + 220 + 311 + 213 + 211 + 210 +
36 + 24 + 22
(89523)2 =
8014367529
[ 74 ] 14_tier
232 + 320 + 227 + 226 + 224 + 315 + 217 + 39 + 213 + 212 + 33 +
33 + 24 + 22
(90144)2 =
8125940736
[ 75 ] 12_tier
232 + 320 + 228 + 226 + 223 + 311 + 310 + 214 + 37 +
210 + 28 + 26
(90153)2 =
8127563409
[ 76 ] 14_tier
232 + 320 + 228 + 226 + 223 + 313 + 218 + 39 + 37 + 28 + 27 +
33 + 33 + 32
(90198)2 =
8135679204
[ 77 ] 11_tier
232 + 320 + 228 + 226 + 224 + 313 + 213 + 211 + 210 + 28 + 27
(91248)2 =
8326197504
[ 78 ] 11_tier
232 + 320 + 229 + 314 + 221 + 219 + 217 + 215 + 38 + 34 + 22
(91605)2 =
8391476025
[ 79 ] 13_tier
232 + 320 + 229 + 226 + 314 + 312 + 218 + 217 + 215 + 212 +
33 + 33 + 23
(92214)2 =
8503421796
[ 80 ] 14_tier
232 + 320 + 229 + 317 + 316 + 223 + 222 + 39 + 38 + 37 + 29 +
28 + 27 + 26
(94695)2 =
8967143025
[ 81 ] 13_tier
233 + 228 + 226 + 225 + 314 + 221 + 220 + 311 + 211 + 210 +
36 + 25 + 22
(95154)2 =
9054283716
[ 82 ] 12_tier
233 + 318 + 226 + 223 + 220 + 218 + 310 + 310 + 211 +
28 + 25 + 32
(96702)2 =
9351276804
[ 83 ] 12_tier
233 + 229 + 227 + 226 + 224 + 314 + 220 + 219 + 213 +
37 + 210 + 28
(97779)2 =
9560732841
[ 84 ] 13_tier
233 + 229 + 318 + 316 + 221 + 220 + 218 + 215 + 214 + 211 +
210 + 33 + 22
(98055)2 =
9614783025
[ 85 ] 14_tier
233 + 229 + 318 + 226 + 224 + 315 + 221 + 311 + 215 + 213 + 38 +
27 + 34 + 24
(98802)2 =
9761835204
[ 86 ] 10_tier
233 + 319 + 223 + 220 + 217 + 216 + 212 + 36 + 29 + 24
(99066)2 =
9814072356
[ 87 ] 13_tier
233 + 319 + 316 + 224 + 313 + 218 + 311 + 214 + 211 + 28 +
33 + 33 + 22

A000187 Prime Curios! Prime Puzzle
Wikipedia 187 Le nombre 187
```

```