A220426 Least nonsquare whose square root starts with at least n even decimal digits.

2, 5, 5, 6, 8, 8, 8, 18, 18, 18, 1652, 2135, 40332, 40332, 78740, 80661, 165389, 165389, 239008, 686015, 4260196, 5018507, 5018507, 5018507, 5018507, 43624023, 43624023, 43624023, 43624023, 43624023, 1801833064, 1801833064, 1801833064, 1801833064, 1801833064

Offset

0,1

Links

Giovanni Resta, Table of n, a(n) for n = 0..43

Example

a(0) = 2 because sqrt(2) = 1.41... starts with 0 even digits, and is the smallest nonsquare for which this is the case.
a(1) = a(2) = 5 because sqrt(5) = 2.23... starts with at least 1 even digit, in fact 2 even digits, whereas sqrt(3) starts off with an odd digit.
a(3) = 6 because sqrt(6) = 2.449... starts off with 3 even digits.

Maple

nexteven:= proc(x)
   local d;
   for d from 0 while x mod 10^(d+1) = 8*(10^(d+1)-1)/9 do od:
   x - 8*(10^(d)-1)/9 + 2*10^(d)
end proc;
evendigits:= proc(x)
local n0, n, d, s;
n0:= ilog10(x);
if type(n0, odd) then n0:= n0-1 end if;
for n from 0 do
   d:= floor(x/10^(n0-2*n));
   s:= floor(sqrt(d));
   while not type(s, integer) do
    Digits:= Digits+2; s:= floor(sqrt(d))
   end do;
   if type(s mod 10, odd) then return n end if;
end do;
end proc;
y:= 2: bsf:= 0: R[0]:= 2:
while bsf < 20 do
   for x from y^2+1 to (y+1)^2-1 do
      v:= evendigits(x);
      if v > bsf then
        for j from bsf+1 to v do R[j]:= x end do;
        bsf:= v;
      end if;
    end do;
   y:= nexteven(y);
end do:
  seq(R[n], n=0..bsf);

Mathematica

f[n_] := Block[{s = Split[ Boole[ EvenQ@# & /@ RealDigits[Sqrt@ n, 10, 32][[1]]]][[1]]}, If[IntegerQ@ Sqrt@ n || Union@ s == {0}, -1, Length@ s]] (* Robert G. Wilson v, Dec 15 2012 *)

Prog

(PARI) a(n) = {my(g=10^(n-1), v); for(k=2, oo, if(setintersect([0, 2, 4, 6, 8], v=Set(digits(floor(sqrt(k)*g))[1..n]))==v && !issquare(k), return(k))); } \\ Jinyuan Wang, Apr 16 2020

Crossrefs

Cf. A210492.

Sequence in context: A130856 A004095 A145434 * A117899 A120839 A332525

Adjacent sequences: A220423 A220424 A220425 * A220427 A220428 A220429

Keyword

nonn,base

Author

Robert Israel, Dec 14 2012

Extensions

a(21)-a(29) from Robert G. Wilson v, Dec 15 2012

a(30)-a(34) from Jinyuan Wang, Apr 16 2020

Status

approved