A290204 Bases b for which there exists an integer y such that y^2 in base b consists of 4 identical digits.

7, 41, 99, 239, 1393, 2943, 8119, 45368, 47321, 82417, 144721, 275807, 470743, 1607521, 9369319, 54608393, 86105599, 184424193, 187869927, 257926007, 318281039, 333815123, 345611082, 500001500, 694220327, 1176320495, 1314587843, 1397186643, 1534997397, 1855077841

Offset

1,1

Comments

Equivalently, numbers k such that A007913(1 + k + k^2 + k^3) < k, where A007913(n) is the squarefree part of n. Sequence is infinite since, as pointed out in Bridy et al., it contains all the terms of A002315 greater than 1. - Giovanni Resta, Jul 25 2017

References

Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit, and Jeffrey Shallit, The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations, Experimental Math, 28 (2019), 428-439.

Links

Table of n, a(n) for n=1..30.

Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit, and Jeffrey Shallit, The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations, preprint arXiv:1707.03894 [math.NT], July 14 2017.

Example

For example, for b = 7, we have y = 20, and the base-b representation of y^2 is 1111.
From Michael De Vlieger, Jul 24 2017: (Start)
Integers y for bases b:
  Base b               Integers y
  ------   ----------------------------------
       7           20,         40
      41         1218
      99         7540
     239        20280,      40560
    1393      1373090
    2943      4903600
    8119     23308460,   46616920
   45368    316540365,  633080730,  949621095,
           1266161460, 1582701825, 1899242190
(End)

Mathematica

Select[Range[2, 3000], Function[b, Count[Map[FromDigits[ConstantArray[#, 4], b] &, Range@ b], k_ /; IntegerQ@ Sqrt@ k] > 0]] (* Michael De Vlieger, Jul 24 2017 *) (* or *)
core[n_] := Block[{f = Transpose@ FactorInteger@ n}, Times @@ (f[[1]]^ Mod[f[[2]], 2])]; Select[Range[10^5], core[1 + # + #^2 + #^3] < # &] (* Giovanni Resta, Jul 25 2017 *)

Crossrefs

Cf. A002315, A007913.

Sequence in context: A031909 A080347 A007398 * A138629 A127923 A230018

Adjacent sequences: A290201 A290202 A290203 * A290205 A290206 A290207

Keyword

nonn,base

Author

Jeffrey Shallit, Jul 24 2017

Extensions

a(16)-a(30) from Giovanni Resta, Jul 25 2017

Status

approved