A273231 Squares that remain squares if you decrease them by 4 times a repunit with the same number of digits.

4, 97344, 462400, 473344, 506944, 846400, 78854400, 444622240000, 448417729600, 454125036544, 551027105344, 824681934400, 983984641600, 460651783840000, 6703941381760000, 444446222224000000, 459134832243732544, 462218702574222400, 462583182938702400

Offset: 1

Author

Paolo P. Lava, May 18 2016

Keywords

• nonn
• easy
• base

Comments

Every term ends in 0 or 4.

Examples

			4 - 4*1 = 0 = 0^2;
97344 - 4*11111 = 52900 = 230^2;
462400 - 4*111111 = 17956 = 134^2.
		

Links

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

Crossrefs

Cf. A002275, A061844, A273299, A273230, A273232-A273234.

Programs

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 *)

Extensions

a(16)-a(19) from Giovanni Resta, May 18 2016