A249853 -id:A249853 - cp's OEIS frontend

A245096 Numbers whose squares become cubes if one of their digits is deleted.

4, 9, 10, 20, 30, 35, 46, 54, 96, 100, 200, 300, 325, 395, 411, 520, 800, 1000, 1470, 2000, 2448, 2700, 3000, 3144, 4000, 4209, 4633, 6400, 6947, 9000, 9051, 10000, 12500, 13719, 20000, 21600, 25300, 30000, 34300, 35000, 46000, 51200, 54000, 61632, 72900, 96000

Offset: 1

Paolo P. Lava, Nov 12 2014

A249853 gives the numbers whose cubes become squares if one of their digits is deleted.

Numbers with single-digit squares are not included. - Davin Park, Dec 30 2016

			4^2 = 16 and (1)^1/3 = 1.
9^2 = 81 and (8)^1/3 = 2 or (1)^1/3 = 1.
10^2 = 100 and (00)^1/3 = 0.
3144^2 = 9884736 and (884736)^1/3 = 96.

Paolo P. Lava, Table of n, a(n) for n = 1..100

Cf. A249853.

with(numtheory): P:=proc(q,h) local a,b,k,n;
for n from 4 to q do a:=n^2; for k from 0 to ilog10(a) do
b:=trunc(a/10^(k+1))*10^k+(a mod 10^k);
if b=trunc(evalf((b)^(1/h)))^h then print(n);
break; fi; od; od; end: P(10^9,3);

f[n_] := !MissingQ@SelectFirst[Delete[IntegerDigits[n^2], #] & /@ Range[IntegerLength[n^2]], IntegerQ@CubeRoot@FromDigits@# &];
Select[Range[4, 1000], f] (* Davin Park, Dec 30 2016 *)
scddQ[x_]:=AnyTrue[Table[FromDigits[Delete[IntegerDigits[x^2],n]],{n, IntegerLength[ x^2]}],IntegerQ[CubeRoot[#]]&]; Select[Range[100000], scddQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 02 2018 *)

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

Original entry on oeis.org

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

Paolo P. Lava, Nov 10 2014

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

			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.

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)

Cf. A248127, A249853.

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);

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 *)

Corrected and extended by Davin Park, Dec 26 2016

Extended by Robert G. Wilson v, Dec 27 2016

A248127 Numbers whose squares became cubes if some digit is prepended, inserted or appended.

Original entry on oeis.org

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

Paolo P. Lava, Nov 10 2014

			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.

Cf. A248051, A249853.

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);

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 *)

Corrected by Davin Park, Dec 30 2016

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A245096 Numbers whose squares become cubes if one of their digits is deleted.

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

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A248127 Numbers whose squares became cubes if some digit is prepended, inserted or appended.

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Programs

Maple

Mathematica

Extensions