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A104266
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Largest n-digit square with no zero digits.
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3
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9, 81, 961, 9216, 99856, 978121, 9998244, 99321156, 999887641, 9978811236, 99999515529, 999332111556, 9999995824729, 99978881115136, 999999961946176, 9999333211115556, 99999999356895225, 999978918111112681, 9999999986285964964, 99999333321111155556
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OFFSET
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1,1
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COMMENTS
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See Formula section for exact formula for terms whose index n is divisible by 4, and upper bounds for other cases; see Links for additional information on those other cases. - Jon E. Schoenfield, Mar 30 2015
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LINKS
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Jon E. Schoenfield, Table of n, a(n) for n = 1..100
Jon E. Schoenfield, Odd-indexed terms with central digits aligned
Jon E. Schoenfield, Patterns and upper bound for terms for which n mod 4 = 2
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FORMULA
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From Jon E. Schoenfield, Mar 31 2015: (Start)
If n is divisible by 4, then a(n) = (10^(n/2) - ceiling(10^(n/4)/3))^2;
otherwise, if n is even, then a(n) < 10^(n) * (1 - (10^-((n-2)/4))* 2 / sqrt(90/1.000000000001026)) (see Links for derivation), except that a(2) = 81.
If n is odd, then a(n) ~ (floor(10^(n/2)))^2. (Although (floor(10*(n/2)))^2 gives an obvious upper bound for a(n) for all n, it seems to be a much tighter upper bound for odd values of n.) (End)
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EXAMPLE
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a(3) = Max{...., 729, 784, 841, 961} = 961.
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MAPLE
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f:= proc(n) local r;
r:= floor(sqrt(10^n));
while has(convert(r^2, base, 10), 0) do r:= r-1 od:
r^2
end proc:
seq(f(n), n=1..24); # Robert Israel, Mar 29 2015
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MATHEMATICA
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f[n_] := Block[{k = Floor[ Sqrt[10^n]]}, While[ Union[ IntegerDigits[ k^2]][[1]] == 0, k-- ]; k^2]; Table[ f[n], {n, 18}] (* Robert G. Wilson v, Mar 03 2005 *)
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PROG
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(PARI) a(n)=k=floor(sqrt(10^n)); while(k, if(type(k)=="t_INT"&&vecmin(digits(k^2)), return(k^2)); k--)
vector(20, n, a(n)) \\ Derek Orr, Mar 29 2015
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CROSSREFS
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Cf. A104265, A104264, A052041.
Sequence in context: A113361 A180737 A068881 * A061433 A069659 A271556
Adjacent sequences: A104263 A104264 A104265 * A104267 A104268 A104269
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KEYWORD
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nonn,base
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AUTHOR
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Reinhard Zumkeller, Feb 26 2005
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EXTENSIONS
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More terms from Robert G. Wilson v, Mar 03 2005
More terms from Jon E. Schoenfield, Mar 29 2015
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STATUS
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approved
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