A273232 Squares that remain squares if you decrease them by 5 times a repunit with the same number of digits.

9, 64, 676, 6084, 56644, 556516, 605284, 669124, 702244, 743044, 784996, 835396, 8538084, 55562116, 60497284, 79673476, 6049417284, 7028810244, 96560590564, 555838838116, 567620600836, 575774404804, 604938617284, 612115334884, 619365852004, 643617898564, 817422124996

Offset

1,1

Comments

Apart from the initial term, any number ends in 4 or 6.

Links

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

Example

9 - 5*1 = 4 = 2^2;
64 - 5*11 = 9 = 3^2;
676 - 5*111 = 121 = 11^2.

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, 5);

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_] := 5 (10^n-1)/9; Flatten[sol /@ r /@ Range[12]] (* Giovanni Resta, May 18 2016 *)

Crossrefs

Cf. A002275, A061844, A273299-A273231, A273233, A273234.

Sequence in context: A037484 A013566 A056769 * A064546 A351591 A217872

Adjacent sequences: A273229 A273230 A273231 * A273233 A273234 A273235

Keyword

nonn,easy,base

Author

Paolo P. Lava, May 18 2016

Status

approved