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A061844 Squares that remain squares if you decrease every digit by 1. 10
1, 36, 3136, 24336, 5973136, 71526293136, 318723477136, 264779654424693136, 24987377153764853136, 31872399155963477136, 58396845218255516736, 517177921565478376336, 252815272791521979771662766736 (list; graph; refs; listen; history; text; internal format)
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

a(79) > 10^209. - JungHwan Min, Jan 02 2016

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

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

Cf. A052382.

Sequence in context: A159728 A268365 A004706 * A036510 A232669 A338076

Adjacent sequences:  A061841 A061842 A061843 * A061845 A061846 A061847

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

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Last modified August 12 00:34 EDT 2022. Contains 356067 sequences. (Running on oeis4.)