A018885 Squares using no more than two distinct digits.

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 225, 400, 441, 484, 676, 900, 1444, 7744, 10000, 11881, 29929, 40000, 44944, 55225, 69696, 90000, 1000000, 4000000, 9000000, 9696996, 100000000, 400000000, 900000000, 6661661161, 10000000000

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• base

Comments

Is 6661661161 the largest term not of the form 10^k, 4*10^k or 9*10^k? Any larger ones must have >= 22 digits. - Robert Israel, Dec 03 2015

Links

• Shawn A. Broyles, Table of n, a(n) for n = 1..85
• Alexandru Gica and Laurentiu Panaitopol, On Oblath's Problem, J. Integer Seqs., Vol. 6(3), 2003, article 03.3.5.
• Eric Weisstein's World of Mathematics, Square Number

Crossrefs

Cf. A016069, A016070, A018884.

Programs

Maple

F:= proc(r, a, b, m)
# get all squares starting with r, with at most m further digits, all from {a,b} where a < b
local res,Ls,Us,L,U,looking;
if issqr(r) then res:= r else res:= NULL fi;
if m = 0 then return res fi;
Ls:= r*10^m + a*(10^m-1)/9;
Us:= r*10^m + b*(10^m-1)/9;
L:= isqrt(Ls);
if L^2 > Ls then L:= L-1 fi;
U:= isqrt(Us);
if U^2 < Us then U:= U+1 fi;
if L > U then res
else res, procname(10*r+a,a,b,m-1), procname(10*r+b,a,b,m-1)
fi
end proc:
S2:= {seq(i^2 mod 100, i=0..99)}:
prs:= map(t -> `if`(t < 10, {0,t},{(t mod 10),(t - (t mod 10))/10}), S2):
prs:= map(p -> `if`(nops(p)=1, seq(p union {s},s={$0..9} minus p), p), prs):
Res:= NULL:
for p in prs do
  a:= min(p); b:= max(p);
  if a > 0 then
     Res:= Res, F(a,a,b,14);
  fi;
  Res:= Res, F(b,a,b,14);
od:
sort(convert({0,Res},list)); # Robert Israel, Dec 03 2015

Mathematica

Select[Range[0, 10^5]^2, Length@ Union@ IntegerDigits@ # <= 2 &] (* Michael De Vlieger, Dec 03 2015 *)
Select[Range[0,100000]^2,Count[DigitCount[#],0]>7&] (* Harvey P. Dale, Jul 25 2020 *)

PARI

for (n=0, 10^6, if ( #Set(digits(n^2))<=2, print1(n^2, ", ") ) ); \\ Michel Marcus, May 21 2015

Formula

For n > 4, a(n) = A016069(n-4)^2.

Extensions

0 inserted and definition edited by Jon E. Schoenfield, Jan 15 2014