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A018885
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Squares using no more than two distinct digits.
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7
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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
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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Alexandru Gica and Laurentiu Panaitopol, On Oblath's Problem, J. Integer Seqs., Vol. 6(3), 2003, article 03.3.5.
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FORMULA
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(PARI) for (n=0, 10^6, if ( #Set(digits(n^2))<=2, print1(n^2, ", ") ) ); \\ Michel Marcus, May 21 2015
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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