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A273231
Squares that remain squares if you decrease them by 4 times a repunit with the same number of digits.
3
4, 97344, 462400, 473344, 506944, 846400, 78854400, 444622240000, 448417729600, 454125036544, 551027105344, 824681934400, 983984641600, 460651783840000, 6703941381760000, 444446222224000000, 459134832243732544, 462218702574222400, 462583182938702400
OFFSET
1,1
COMMENTS
Every term ends in 0 or 4.
LINKS
EXAMPLE
4 - 4*1 = 0 = 0^2;
97344 - 4*11111 = 52900 = 230^2;
462400 - 4*111111 = 17956 = 134^2.
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, 4);
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_] := 4 (10^n-1)/9; Flatten[sol /@ r /@ Range[12]] (* Giovanni Resta, May 18 2016 *)
KEYWORD
nonn,easy,base
AUTHOR
Paolo P. Lava, May 18 2016
EXTENSIONS
a(16)-a(19) from Giovanni Resta, May 18 2016
STATUS
approved