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

1,1

Comments

Any number ends in 9.

Links

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

Example

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

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, 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 *)

Crossrefs

Cf. A002275, A061844, A273299-A273233.

Sequence in context: A133414 A013850 A174001 * A259159 A340181 A131678

Adjacent sequences: A273231 A273232 A273233 * A273235 A273236 A273237

Keyword

nonn,easy,base

Author

Paolo P. Lava, May 18 2016

Extensions

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

Status

approved