A289390 Integers b for which there exists an integer y < b such that b^2+1 divides y^4.

239, 682, 4443, 12943, 275807, 6826318, 26392464, 30349818, 54608393, 54610269, 103224943, 275805068, 419282318, 1085592682, 1268860318, 1344783432, 2321201748

Offset

1,1

Comments

Equivalently, bases b for which there exists an integer y such that y^4 in base b looks like [c,d,c,d] for some base-b digits c, d.

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..17.

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 = 239, we have y = 78 with 239^2+1 = 2*13^4 dividing 78^4 = 2^4*3^4*13^4. Equivalently, the base-b representation of y^4 is (2,170,2,170).

Mathematica

Select[Range[300000], Times @@ Table[ f[[1]]^(3 - Mod[f[[2]] - 1, 4]), {f, FactorInteger[1 + #^2]}] <= #^2 + 1 &] (* Giovanni Resta, Jul 26 2017 *)

Crossrefs

Cf. A290204.

Sequence in context: A164290 A201787 A390000 * A118574 A142854 A065098

Adjacent sequences: A289387 A289388 A289389 * A289391 A289392 A289393

Keyword

nonn,more

Author

Jeffrey Shallit, Jul 25 2017

Extensions

a(9)-a(17) from Giovanni Resta, Jul 26 2017

Better definition added, base keyword removed by Max Alekseyev, Oct 28 2025

Status

approved