OFFSET
1,2
COMMENTS
The terms may be calculated efficiently by solving x^2 - y^2 = 111...1; this is done by factoring 111..1 = (x + y)(x - y).
Note that some solutions will produce a square containing a zero digit so the solution is impermissible; for example, 460^2 - 317^2 = 111111 but 460^2 = 211600. - Wendy Appleby, Sep 20 2015
Except for a(1) = 1, we don't allow decreasing the digits to create a leading 0. Thus 126736 = 356^2 is not included, even though 126736 - 111111 = 15625 = 125^2. - Robert Israel, Dec 30 2015
If it exists, a(79) > 10^262. - Max Alekseyev, Sep 05 2023
From Robert Israel, Jan 04 2016: (Start)
The sequence may well be finite.
It is known that A000005(n) = O(n^epsilon) for all epsilon>0.
Therefore if 1 < c < 10/9, for d sufficiently large (10^d-1)/9 has fewer than c^d divisors, and thus fewer than c^d possible candidates for x^2 having d digits.
Heuristically, x^2 has probability ~ (9/10)^d of having no digits 0.
Thus we expect fewer than (9c/10)^d terms having d digits.
Since Sum_d (9c/10)^d converges, we expect only finitely many terms.
Of course, this is only a heuristic argument, but it seems to fit well with the data. (End)
LINKS
JungHwan Min, Table of n, a(n) for n = 1..78
FORMULA
EXAMPLE
13225 = 115^2 and 24336 = 156^2.
MAPLE
A:= {1}:
for d from 1 to 96 do
r:= (10^d-1)/9;
f:= subs(X=10, factors((X^d-1)/(X-1))[2]);
q:= map(t -> op(map(s -> [s[1], t[2]*s[2]], ifactors(t[1])[2])), f);
divs:= {1};
for t in q do
divs:= map(x -> seq(x*t[1]^j, j=0..t[2]), divs)
od;
for t in select(s -> s^2 > r, divs) do
x:= (t + r/t)/2;
if ilog10(x^2) = d-1 and x^2 > 2*10^(d-1) and not has(convert(x^2, base, 10), 0) then
A:= A union {x^2};
fi
od
od:
sort(convert(A, list)); # Robert Israel, Dec 30 2015
MATHEMATICA
For[digits = 1, digits <= 30, digits++, n = (10^digits - 1)/9; divList = Select[Divisors[n], (#1 >= Sqrt[n])&]; For[j = 1, j <= Length[divList], j++, x = (divList[[j]] + n/divList[[j]])/2; y = (divList[[j]] - n/divList[[j]])/2; dx = IntegerDigits[x^2]; dy = IntegerDigits[y^2]; If[(Length[dx] == digits) && (Length[dy] == digits) && (Select[dx, (#1 == 0)&] == {}), Print[x^2]]]]
Flatten@Prepend[Table[Select[#[[Ceiling[(Length[#] + 1)/2] ;; ]] &@(# + Reverse@#)/2 &@Divisors[(10^n - 1)/9], IntegerLength[#^2] == n && (#[[1]] != 1 && FreeQ[#, 0]&[IntegerDigits[#^2]])&]^2, {n, 30}], 1] (* JungHwan Min, Dec 29 2015 *)
Join[{1}, Select[Select[Flatten[Table[#^2&/@(x/.Solve[{x^2-y^2 == FromDigits[ PadRight[{}, n, 1]], x>0, y>0}, {x, y}, Integers]), {n, 2, 30}]], DigitCount[ #, 10, 0]==0&&IntegerDigits[#][[1]]>1&]// Union, IntegerQ[ Sqrt[ FromDigits[IntegerDigits[#]-1]]]&]] (* Harvey P. Dale, Apr 16 2016 *)
CROSSREFS
KEYWORD
base,nonn,nice
AUTHOR
Erich Friedman, Jun 23 2001
EXTENSIONS
More terms and program from Jonathan Cross (jcross(AT)wcox.com), Oct 08 2001
STATUS
approved