A248051 Numbers whose cubes become squares if some digit is prepended, inserted or appended.

1, 2, 5, 6, 10, 25, 30, 40, 41, 60, 84, 90, 96, 100, 121, 129, 160, 169, 200, 201, 250, 266, 360, 400, 490, 500, 600, 640, 724, 810, 1000, 1025, 1210, 1440, 1690, 1960, 2250, 2500, 2560, 2890, 3000, 3240, 3604, 3610, 4000, 4100, 4410, 4840, 5216, 5290, 5760

Offset

1,2

Comments

No leading zeros allowed.

Number of terms <= 10^k for k = 0, 1, 2, ...: 1, 5, 14, 31, 64, 144, 373, ..., . Robert G. Wilson v, Dec 27 2016

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..373 (terms 1..100 from Paolo P. Lava, terms 101..144 from Davin Park)

Example

If n = 1 then n^3 = 1 and if we append a 6 we have sqrt(16) = 4.
If n = 2 then n^3 = 8 and if we append a 1 we have sqrt(81) = 9.
If n = 5 then n^3 = 125 and if we insert a 2 we get sqrt(1225) = 35.
Again, if n = 25 then n^3 = 15625 and we have sqrt(105625) = 325 or sqrt(156025) = 395.

Maple

with(numtheory): P:=proc(q) local a, b, j, k, n, ok;
for n from 1 to q do a:=n^3; b:=ilog10(a)+1; ok:=1;
for k from 0 to b do if ok=1 then for j from 0 to 9 do
if not (j=0 and k=b) then if type(sqrt(trunc(a/10^k)*10^(k+1)+j*10^k+(a mod 10^k)), integer)
then print(n); ok:=0; break; fi; fi; od; fi;
od; od; end: P(10^6);

Mathematica

f[n_] := ! MissingQ@SelectFirst[Rest@Flatten[Outer[Insert[IntegerDigits[n^3], #2, #1] &, Range[IntegerLength[n^3] + 1], Range[0, 9]], 1], IntegerQ@Sqrt@FromDigits@# &];
Select[Range[100], f] (* Davin Park, Dec 28 2016 *)

Crossrefs

Cf. A248127, A249853.

Sequence in context: A123645 A057252 A057250 * A056643 A057256 A236248

Adjacent sequences: A248048 A248049 A248050 * A248052 A248053 A248054

Keyword

nonn,base

Author

Paolo P. Lava, Nov 10 2014

Extensions

Corrected and extended by Davin Park, Dec 26 2016

Extended by Robert G. Wilson v, Dec 27 2016

Status

approved