The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

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