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

Author

Paolo P. Lava, May 18 2016

Keywords

• nonn
• easy
• base

Comments

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

Examples

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

Links

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

Crossrefs

Cf. A002275, A061844, A273229, A273231-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(10^9,3);

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 *)