A132391 Numbers whose square starts with 4 identical digits.

2357, 2582, 3334, 4714, 5774, 6667, 8165, 8819, 9428, 10541, 10542, 10543, 10544, 10545, 14907, 14908, 14909, 18257, 18258, 18259, 21081, 21082, 21083, 23570, 23571, 25819, 25820, 27888, 27889, 29813, 29814, 31622, 33332, 33333

Offset: 1

Author

Jonathan Vos Post, Aug 29 2007

Keywords

• nonn
• base
• look

Examples

			Example: 2357^2 = 5555449.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A119887, A119511, A131573, A119866, A133183.

Programs

Maple

R:= NULL: count:= 0:
for d from 1 while count < 100 do
  for i from 1 to 9 do
    L:= i*1111*10^d;
    X:= [$ceil(sqrt(L)) .. floor(sqrt(L+10^d-1))];
    m:= nops(X);
    if m > 0 then
      count:= count+nops(X);
      R:= R, op(X);
    fi
od od:
R; # Robert Israel, Mar 12 2021

Mathematica

Select[Range[10, 50000], Length[Union[Take[IntegerDigits[ #^2], 4]]] == 1 & ]
(* or *)
(* Here's a more generic Mathematica program that calculates the first q terms of squares starting with n identical digits *)
n=4; q=30; t=Table[(10^n-1)*i/9, {i,1,9}]; u=Sqrt[Union[t,10*t]];
v=Sqrt[Union[t+1, 10*(t+1)]]; k=1; While[s=Sort[Flatten[Table[Union
[Table[Range[Ceiling[10^j*u[[i]]], f=10^j*v[[i]]; If[IntegerQ[f],
f=f-1]; Floor[f]], {i,1,18}]], {j,0,k}]]]; Length[s]Hans Havermann, Aug 30 2007 *)

Python

def aupto(limit):
  alst = []
  for m in range(34, limit+1):
    if len(set(str(m*m)[:4])) == 1: alst.append(m)
  return alst
print(aupto(33333)) # Michael S. Branicky, Mar 12 2021