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A104265 Smallest n-digit square with no zero digits.

%I A104265 #24 Jun 12 2020 14:08:13
%S A104265 1,16,121,1156,11236,111556,1115136,11115556,111112681,1111155556,
%T A104265 11111478921,111111555556,1111118377216,11111115555556,
%U A104265 111111226346761,1111111155555556,11111112515384644,111111111555555556,1111111112398242916,11111111115555555556,111111111113333185156,1111111111155555555556
%N A104265 Smallest n-digit square with no zero digits.
%H A104265 Chai Wah Wu, <a href="/A104265/b104265.txt">Table of n, a(n) for n = 1..362</a>
%F A104265 From _Chai Wah Wu_, Mar 24 2020: (Start)
%F A104265 a(n) >= (10^n-1)/9.
%F A104265 a(2n) = (10^n+2)^2/9 = A102807(n). Proof: the smallest 2n-digit number without zero digits is (10^(2n)-1)/9. ((10^n-1)/3)^2 = (10^(2n)-2*10^n+1)/9 < (10^(2n)-1)/9 for n >= 1. Thus a(2n) > ((10^n-1)/3)^2. The next square is ((10^n+2)/3)^2 = (10^(2n)-1)/9 + 4*(10^(n)-1)/9 + 1, i.e. it is n 1's followed by n-1 5's followed by the digit 6, and has no zero digits.
%F A104265 (End)
%e A104265 a(3) = Min{121, 144, 169, 196, ....} = 121.
%t A104265 f[n_] := Block[{k = Ceiling[ Sqrt[10^n]]}, While[ Union[ IntegerDigits[ k^2]][[1]] == 0, k++ ]; k^2]; Table[ f[n], {n, 0, 20}] (* _Robert G. Wilson v_, Mar 02 2005 *)
%t A104265 snds[n_]:=Module[{c=Ceiling[Sqrt[FromDigits[Join[PadRight[{},n-1,1], {0}]]]]^2},While[DigitCount[c,10,0]>0,c=(1+Sqrt[c])^2];c]; Array[ snds,22] (* _Harvey P. Dale_, Jun 12 2020 *)
%o A104265 (Python)
%o A104265 from sympy import integer_nthroot
%o A104265 def A104265(n):
%o A104265     m, a = integer_nthroot((10**n-1)//9,2)
%o A104265     if not a:
%o A104265         m += 1
%o A104265     k = m**2
%o A104265     while '0' in str(k):
%o A104265         m += 1
%o A104265         k += 2*m-1
%o A104265     return k # _Chai Wah Wu_, Mar 24 2020
%Y A104265 Cf. A102807, A104266, A104264, A052041.
%K A104265 nonn,base
%O A104265 1,2
%A A104265 _Reinhard Zumkeller_, Feb 26 2005
%E A104265 More terms from _Robert G. Wilson v_, Mar 02 2005
%E A104265 Two more terms from _Jon E. Schoenfield_, Mar 29 2015
%E A104265 a(21)-a(22) from _Chai Wah Wu_, Mar 24 2020