A273229 - OEIS

%I #9 May 21 2016 22:42:13

%S 1,36,400,3136,24336,115600,118336,126736,211600,309136,430336,577600,

%T 5973136,19713600,30869136,53582400,3086469136,4310710336,71526293136,

%U 111155560000,112104432400,113531259136,137756776336,206170483600,245996160400,262303768336,308642469136

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

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

%H Giovanni Resta, <a href="/A273229/b273229.txt">Table of n, a(n) for n = 1..10000</a>

%e 1 - 1 = 0 = 0^2;

%e 36 - 11 = 25 = 5^2;

%e 400 - 111 = 289 = 17^2;

%p P:=proc(q,h) local n; for n from 1 to q do

%p if type(sqrt(n^2-h*(10^(ilog10(n^2)+1)-1)/9),integer) then print(n^2);

%p fi; od; end: P(10^9,1);

%t 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 *)

%Y Cf. A002275, A061844, A273230-A273234.

%K nonn,easy,base

%O 1,2

%A _Paolo P. Lava_, May 18 2016