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A273229
Squares that remain squares if you decrease them by a repunit with the same number of digits.
3
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
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
KEYWORD
nonn,easy,base
AUTHOR
Paolo P. Lava, May 18 2016
STATUS
approved