A273234 Squares that remain squares if you decrease them by 8 times a repunit with the same number of digits.

9, 889249, 896809, 908209, 902942754289, 924745719769, 946618081249, 987107822089, 910909843526089, 9810767198166489, 888909576913320169, 889214944824055249, 889286612895723249, 889972999762742809, 890923059538260849, 896642235371330809, 896979367708462809

Offset: 1

Author

Paolo P. Lava, May 18 2016

Keywords

• nonn
• easy
• base

Comments

Any number ends in 9.

Examples

			9 - 8*1 = 1 = 1^2;
889249 - 8*111111 = 361 = 19^2;
896809 - 8*111111 = 7921 = 89^2.
		

Links

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

Crossrefs

Cf. A002275, A061844, A273299-A273233.

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(10^9,8);

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_] := 8 (10^n-1)/9; Flatten[sol /@ r /@ Range[12]] (* Giovanni Resta, May 18 2016 *)

Extensions

a(11)-a(17) from Giovanni Resta, May 18 2016