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

1,1

Comments

Any number ends in 1.

Links

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

Example

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

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, 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 *)

Crossrefs

Cf. A002275, A061844, A273299-A273232, A273234.

Sequence in context: A224002 A253461 A247842 * A045792 A067478 A076749

Adjacent sequences: A273230 A273231 A273232 * A273234 A273235 A273236

Keyword

nonn,easy,base

Author

Paolo P. Lava, May 18 2016

Status

approved