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
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
KEYWORD
nonn,base
AUTHOR
Jeffrey Shallit, Jul 24 2017
EXTENSIONS
a(16)-a(30) from Giovanni Resta, Jul 25 2017
STATUS
approved