A248127 Numbers whose squares became cubes if some digit is prepended, inserted or appended.
2, 4, 5, 10, 31, 72, 75, 80, 162, 270, 383, 640, 1250, 2000, 2160, 3430, 4000, 5000, 5120, 7290, 10000, 13310, 17280, 21970, 27440, 28875, 31000, 33750, 40960, 49130, 58320, 68590, 72000, 75000, 80000, 92610
Offset: 1
Examples
If n = 10 then n^2 = 100 and if we append a 0 we have (1000)^1/3 = 10. If n = 31 then n^2 = 961 and if we insert a 2 we have (9261)^1/3 = 21. Again, if n = 112625 then n^2 = 12684390625 and if we insert an 8 we have (126884390625)^1/3 = 5025.
Programs
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Maple
with(numtheory): P:=proc(q) local a, b, c, j, k, ok, n; for n from 1 to q do a:=n^2; c:=ilog10(a)+1; ok:=1; for k from 0 to ilog10(a)+1 do if ok=1 then for j from 0 to 9 do if not (k=c and j=0) then b:=trunc(a/10^k)*10^(k+1)+j*10^k+(a mod 10^k); if b=trunc(evalf((b)^(1/3)))^3 then print(n); ok:=0; break; fi; fi; od; fi; od; od; end: P(10^9);
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Mathematica
f[n_] := ! MissingQ@SelectFirst[Rest@Flatten[Outer[Insert[IntegerDigits[n^2], #2, #1] &, Range[IntegerLength[n^2] + 1], Range[0, 9]], 1], IntegerQ@CubeRoot@FromDigits@# &]; Select[Range[100], f] (* Davin Park, Dec 30 2016 *)
Extensions
Corrected by Davin Park, Dec 30 2016