A290173 Bases b for which there exists an integer y such that y^2 in base b looks like [c,d,c,d,c,d] for some base-b digits c, d.

68, 313, 699, 4366, 51567, 234924, 686287, 3526450, 3652434, 301121223, 1250094151, 3072665429, 4507947478

Offset

1,1

Comments

Equivalently, numbers k such that A007913(1 + k^2 + k^4) < k^2, where A007913(n) is the squarefree part of n. - Giovanni Resta, Jul 27 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..13.

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 = 68, we have y = 160797, and the base-b representation of y^2 is (17,53,17,53,17,53).

Mathematica

core[n_] := Block[{f = Transpose@ FactorInteger@ n}, Times @@ (f[[1]]^ Mod[f[[2]], 2])]; Select[Range[3*10^5], core[1 + # + #^2] core[1 - # + #^2] < #^2 &] (* Giovanni Resta, Jul 27 2017 *)

Prog

(PARI) isok(k) = core(1+k^2+k^4) < k^2; \\ Michel Marcus, Jul 28 2017

Crossrefs

Cf. A290172, A290176, A290177, A290185, A007913.

Sequence in context: A200198 A237745 A237740 * A281770 A032510 A211691

Adjacent sequences: A290170 A290171 A290172 * A290174 A290175 A290176

Keyword

nonn,base,more

Author

Jeffrey Shallit, Jul 23 2017

Extensions

a(10)-a(13) from Giovanni Resta, Jul 27 2017

Status

approved