A273230 Squares that remain squares if you decrease them by 3 times a repunit with the same number of digits.

4, 49, 529, 4489, 38809, 344569, 363609, 375769, 444889, 558009, 597529, 700569, 7198489, 35366809, 44448889, 65983129, 4444488889, 5587114009, 83574762649, 335330171929, 359763638809, 390241344249, 403831017529, 407200963129, 435775577689, 444444888889, 453557800089

Offset

1,1

Comments

Apart from the initial term, any number ends in 9.

Links

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

Example

4 - 3*1 = 1 = 1^2;
49 - 3*11 = 16 = 4^2;
529 - 3*111 = 196 = 14^2.

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, 3);

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_] := 3 (10^n-1)/9; Flatten[sol /@ r /@ Range[12]] (* Giovanni Resta, May 18 2016 *)

Crossrefs

Cf. A002275, A061844, A273229, A273231-A273234.

Sequence in context: A198971 A348547 A348425 * A199028 A189146 A086094

Adjacent sequences: A273227 A273228 A273229 * A273231 A273232 A273233

Keyword

nonn,easy,base

Author

Paolo P. Lava, May 18 2016

Status

approved