A104266 Largest n-digit square with no zero digits.

9, 81, 961, 9216, 99856, 978121, 9998244, 99321156, 999887641, 9978811236, 99999515529, 999332111556, 9999995824729, 99978881115136, 999999961946176, 9999333211115556, 99999999356895225, 999978918111112681, 9999999986285964964, 99999333321111155556

Offset: 1

Author

Reinhard Zumkeller, Feb 26 2005

Keywords

• nonn
• base

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

Examples

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

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

Crossrefs

Cf. A104265, A104264, A052041.

Programs

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 *)

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

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)

Extensions

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

More terms from Jon E. Schoenfield, Mar 29 2015