A273233 Squares that remain squares if you decrease them by 7 times a repunit with the same number of digits.

81, 841, 7921, 77841, 790321, 863041, 982081, 9991921, 79014321, 80299521, 94653441, 7901254321, 8635799041, 778133930161, 790123654321, 794396081521, 816057482881, 965485073281, 989863816561, 79012347654321, 86358529399041, 857789228465521, 7901234587654321, 8547733055510401

Offset: 1

Author

Paolo P. Lava, May 18 2016

Keywords

• nonn
• easy
• base

Comments

Any number ends in 1.

Examples

			81 - 7*11 = 4 = 2^2;
841 - 7*111 = 64 = 8^2;
7921 - 7*1111 = 144 = 12^2.
		

Links

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

Crossrefs

Cf. A002275, A061844, A273299-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(10^9,7);

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_] := 7 (10^n-1)/9; Flatten[sol /@ r /@ Range[12]] (* Giovanni Resta, May 18 2016 *)