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A104266 Largest n-digit square with no zero digits. 3
9, 81, 961, 9216, 99856, 978121, 9998244, 99321156, 999887641, 9978811236, 99999515529, 999332111556, 9999995824729, 99978881115136, 999999961946176, 9999333211115556, 99999999356895225, 999978918111112681, 9999999986285964964, 99999333321111155556 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

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

FORMULA

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)

EXAMPLE

a(3) = Max{...., 729, 784, 841, 961} = 961.

MAPLE

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

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A104265, A104264, A052041.

Sequence in context: A113361 A180737 A068881 * A061433 A069659 A271556

Adjacent sequences:  A104263 A104264 A104265 * A104267 A104268 A104269

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Feb 26 2005

EXTENSIONS

More terms from Robert G. Wilson v, Mar 03 2005

More terms from Jon E. Schoenfield, Mar 29 2015

STATUS

approved

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Last modified October 22 01:48 EDT 2021. Contains 348160 sequences. (Running on oeis4.)