A273229 Squares that remain squares if you decrease them by a repunit with the same number of digits.

1, 36, 400, 3136, 24336, 115600, 118336, 126736, 211600, 309136, 430336, 577600, 5973136, 19713600, 30869136, 53582400, 3086469136, 4310710336, 71526293136, 111155560000, 112104432400, 113531259136, 137756776336, 206170483600, 245996160400, 262303768336, 308642469136

Offset

1,2

Comments

Apart from the initial term, any number ends in 0 or 6.

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000

Example

1 - 1 = 0 = 0^2;
36 - 11 = 25 = 5^2;
400 - 111 = 289 = 17^2;

Maple

P:=proc(q, h) local n; for n from 1 to q do
if type(sqrt(n^2-h*(10^(ilog10(n^2)+1)-1)/9), integer) then print(n^2);
fi; od; end: P(10^9, 1);

Mathematica

sol[k_] := Block[{x, e = IntegerLength@k, d = Divisors@k}, Union[#+k/# & /@ Select[ Take[d, Ceiling[ Length@d/2]], EvenQ[ x= #+k/#] && IntegerLength[ x^2/4] == e &]]^2/4]; r[n_] := (10^n-1)/9; Flatten[sol /@ r /@ Range[12]] (* Giovanni Resta, May 18 2016 *)

Crossrefs

Cf. A002275, A061844, A273230-A273234.

Sequence in context: A086575 A203027 A164583 * A264361 A055862 A105095

Adjacent sequences: A273226 A273227 A273228 * A273230 A273231 A273232

Keyword

nonn,easy,base

Author

Paolo P. Lava, May 18 2016

Status

approved