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A290204 Bases b for which there exists an integer y such that y^2 in base b consists of 4 identical digits. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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: A105902 A031909 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

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Last modified May 31 00:53 EDT 2020. Contains 334747 sequences. (Running on oeis4.)