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

Author

Paolo P. Lava, Nov 10 2014

Keywords

• nonn
• base

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.
		

Crossrefs

Cf. A248051, A249853.

Programs

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);

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