A290177 Bases b for which there exists an integer y such that y^3 in base b consists of three identical digits.

18, 88916, 1147805, 6042955, 761357755, 1161183643

Offset

1,1

Comments

A number b is a term if 1+b+b^2 can be multiplied by a number k < b to obtain a cube. The smallest candidate k can be easily obtained from the prime factorization of 1+b+b^2. For example, for b = 761357755 we have 1+b+b^2 = 3 * 7^2 * 2131 * 12277^3, so k = 3^2 * 7 * 2131^2, which happens to smaller than b. a(7) > 10^10. - Giovanni Resta, Sep 02 2019

From Bert Dobbelaere, Apr 07 2026: (Start)

If b is a term, then b^2+b+1 cannot be cubefree. We can factor it uniquely as b^2+b+1 = u * v^2 * w^3 where u*v is squarefree. Then b is a term iff k = u^2*v < b. Suppose w=1, then k^2 < b^2 < b^2+b+1 would imply u^4*v^2 < u*v^2, which is impossible for positive u and v.

a(7) > 2*10^11. (End)

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..6.

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 = 18, we have y = 7, and the base-b representation of y^3 is 111.

Crossrefs

Cf. A290172, A290173, A290176, A290185.

Sequence in context: A051591 A146203 A192081 * A259364 A013762 A078352

Adjacent sequences: A290174 A290175 A290176 * A290178 A290179 A290180

Keyword

nonn,base,more

Author

Jeffrey Shallit, Jul 23 2017

Extensions

a(5)-a(6) from Giovanni Resta, Sep 02 2019

Status

approved