10   dim Typ(6),Stval(3,4),Endval(3,4),V(3)
   20   Typ(1)="spc":Typ(2)="scp"
   30   Typ(3)="psc":Typ(4)="pcs"
   40   Typ(5)="cps":Typ(6)="csp"
   45   open "num-meld.txt" for output as #2
   50 
   60   ' squares 4-9 are 2-digit
   70   ' squares 10-31 are 3-digit
   80   ' square 32 is first 4-dig square; 99th is last
   90   ' primes 5-25 are 2-digit
  100   ' primes 26-168 are 3-digit
  110   ' prime 169 is first 4-dig prime; 1229th is last
  120   ' cubes 3-4 are 2-digit
  130   ' cubes 5-9 are 3-digit
  140   ' cube 10 is first 4-dig cube; 21st is last
  150   Stval(1,2)=4:Endval(1,2)=9
  160   Stval(1,3)=10:Endval(1,3)=31
  170   Stval(1,4)=32:Endval(1,4)=99
  180   Stval(2,2)=5:Endval(2,2)=25
  190   Stval(2,3)=26:Endval(2,3)=168
  200   Stval(2,4)=169:Endval(2,4)=1229
  210   Stval(3,2)=3:Endval(3,2)=4
  220   Stval(3,3)=5:Endval(3,3)=9
  230   Stval(3,4)=10:Endval(3,4)=21
  240 
  250 
  260   for Digs=2 to 4
  270    print #2,:print #2,"digits: ";Digs:print #2,
  280    for Tp=1 to 6
  290      Proto=Typ(Tp)
  300      print #2,Proto
  310      if left(Proto,1)="s" then St=Stval(1,Digs):Fin=Endval(1,Digs)
  320      if left(Proto,1)="p" then St=Stval(2,Digs):Fin=Endval(2,Digs)
  330      if left(Proto,1)="c" then St=Stval(3,Digs):Fin=Endval(3,Digs)
  340      for Bse=St to Fin
  350        if left(Proto,1)="s" then V(1)=cutspc(str(Bse*Bse))
  360        if left(Proto,1)="p" then V(1)=cutspc(str(prm(Bse)))
  370        if left(Proto,1)="c" then V(1)=cutspc(str(Bse*Bse*Bse))
  380        for Varpos=1 to Digs
  390         for Substdig=0 to 9
  400          Good=1
  410          if Substdig=0 and Varpos=1 then Good=0
  420          if Substdig=0 or val(mid(V(1),Varpos,1))=Substdig then Good=0
  430          Newv=left(V(1),Varpos-1)+cutspc(str(Substdig))+mid(V(1),Varpos+1,*)
  440          Newval=val(Newv)
  450          if mid(Proto,2,1)="s" then if fnIsSq(Newval)=0 then Good=0
  460          if mid(Proto,2,1)="p" then if fnIsPrime(Newval)=0 then Good=0
  470          if mid(Proto,2,1)="c" then if fnIsCube(Newval)=0 then Good=0
  480       
  490          if Good then
  500           :V(2)=Newv
  510           :for Varpos2=1 to Digs
  520            :for Subdig2=0 to 9
  530             :Good=1
  540             :if Subdig2=0 and Varpos2=1 then Good=0:endif
  550             :if Subdig2=0 or val(mid(V(2),Varpos2,1))=Subdig2 then Good=0:endif
  560             :Newv=left(V(2),Varpos2-1)+cutspc(str(Subdig2))+mid(V(2),Varpos2+1,*)
  570             :Newval=val(Newv)
  580             :if mid(Proto,3,1)="s" then if fnIsSq(Newval)=0 then Good=0:endif:endif
  590             :if mid(Proto,3,1)="p" then if fnIsPrime(Newval)=0 then Good=0:endif:endif
  600             :if mid(Proto,3,1)="c" then if fnIsCube(Newval)=0 then Good=0:endif:endif
  610             :
  620             :if Good then
  630              :V(3)=Newv
  640              :for I=1 to 3
  641                :print #2,V(I);" ";:Dup=0
  642                :for J=1 to len(V(I))-1:if instr(mid(V(I),J+1,*),mid(V(I),J,1))>0 then Dup=1:endif:next
  643                :if Dup then print #2,"* ";:endif
  644              :next
  650              :print #2,
  660             :endif
  670            :next Subdig2
  680           :next Varpos2
  690       
  700         next Substdig
  710        next Varpos
  720      next
  730    next Tp
  740   next Digs
  750   close
  998   end
  999 
 1000   fnIsPrime(X)
 1010   local A
 1020   if X=prmdiv(X) and X<>1 then A=1 else A=0
 1030   return(A)
 1040 
 2000   fnIsSq(X)
 2010   local A,Sr
 2020   Sr=int(sqrt(X)+0.5)
 2030   if Sr*Sr=X then A=1 else A=0
 2040   return(A)
 2050 
 3000   fnIsCube(X)
 3010   local A,Cr
 3020   Cr=int(X^(1/3)+0.5)
 3030   if Cr*Cr*Cr=X then A=1 else A=0
 3040   return(A)

find solutions including those with duplicate digits within numbers, and it marks those with asterisks.

The cleaned up output file (line with asterisks removed) is as follows. The heading abbreviations are spc for square prime cube, etc.:

digits:  2

spc
16 17 27
25 23 27
25 29 27
36 37 27
49 29 27
49 47 27
64 61 64
64 67 27
64 67 64
81 61 64
scp
25 27 17
25 27 37
25 27 47
25 27 67
25 27 97
25 27 23
25 27 29
psc
23 25 27
29 25 27
pcs
17 27 25
23 27 25
29 27 25
37 27 25
47 27 25
67 27 25
97 27 25
cps
27 17 16
27 37 36
27 47 49
27 67 64
27 23 25
27 29 49
27 29 25
64 61 81
64 61 64
64 67 64
csp
27 25 23
27 25 29

digits:  3

spc
169 769 729
529 829 729
729 829 729
729 719 729
729 739 729
729 769 729
scp
529 729 829
529 729 719
529 729 739
529 729 769
625 125 127
psc
251 256 216
257 256 216
509 529 729
521 529 729
523 529 729
569 529 729
829 529 729
pcs
127 125 625
709 729 529
719 729 529
739 729 529
769 729 529
829 729 529
cps
729 829 529
729 829 729
729 719 729
729 739 729
729 769 169
729 769 729
csp
216 256 251
216 256 257
729 529 829
729 529 569
729 529 521
729 529 523

digits:  4

spc
4096 4091 4096
4096 4093 4096
4761 4261 9261
6241 9241 9261
scp
psc
pcs
cps
4096 4091 4096
4096 4093 4096
9261 4261 4761
9261 9241 6241
csp

Only the 4-digit case has categories lacking solutions, so I'm appending the portion of the full output for just those categories within the 4-digit set. I've deleted the asterisks.:

scp
2704 2744 2741
2704 2744 2749
6889 6859 9859
6889 6859 6359
6889 6859 6659
6889 6859 6959
6889 6859 6829
6889 6859 6869
6889 6859 6899
6889 6859 6857
7744 2744 2741
7744 2744 2749
psc
1889 6889 6859
2707 2704 2744
3889 6889 6859
4889 6889 6859
6089 6889 6859
6389 6889 6859
6689 6889 6859
6829 6889 6859
6869 6889 6859
6883 6889 6859
6899 6889 6859
7741 7744 2744
pcs
2741 2744 7744
2749 2744 7744
6359 6859 6889
6659 6859 6889
6829 6859 6889
6857 6859 6889
6869 6859 6889
6899 6859 6889
6959 6859 6889
9859 6859 6889

csp
1000 1600 1601
1000 1600 1607
1000 1600 1609
2744 7744 7741
6859 6889 1889
6859 6889 3889
6859 6889 4889
6859 6889 6389
6859 6889 6689
6859 6889 6829
6859 6889 6869
6859 6889 6899
6859 6889 6883
8000 8100 8101

The above do not all validate that the entries be in ascending order. I reran with a check for that, and the results are shown below, with a mixture of solutions that do and do not have duplicate digits within words. Numbers with duplicate digits are marked with an asterisk for ease in spotting them:

digits:  2

spc
16 17 27
scp
25 27 37
25 27 47
25 27 67
25 27 97
25 27 29
psc
23 25 27
pcs
cps
27 47 49
27 29 49
csp

digits:  3

spc
225 * 229 * 729
scp
121 * 125 127
529 729 829
529 729 929 *
529 729 739
529 729 769
psc
101 * 121 * 125
229 * 529 729
509 529 729
521 529 729
523 529 729
pcs
211 * 216 256
cps
csp
125 225 * 227 *
125 225 * 229 *
216 256 257

digits:  4

spc
1521 * 1721 * 1728
4900 * 4903 4913
6241 9241 9261
scp
2704 2744 * 2749
psc
pcs
2741 2744 * 7744 *
6359 6859 6889 *
6659 * 6859 6889 *
6829 6859 6889 *
6857 6859 6889 *
cps
1000 * 1009 * 1089
1331 * 1931 * 1936
1331 * 1361 * 1369
1331 * 1381 * 1681 *
6859 6869 * 6889 *
csp
1000 * 1600 * 1601 *
1000 * 1600 * 1607
1000 * 1600 * 1609
6859 6889 * 6899 *
8000 * 8100 * 8101 *

Note that only one 4-digit sequence follows all the rules: the square-prime-cube sequence 6241 9241 9261.