A118020 Number of steps to reach an integer starting with (n+1)/n and using the approximate cubing map x -> x*ceiling(x^2); or -1 if no integer is ever reached.

0, -1, 3, -1, 12, 2, 3, -1, 7, 2, 18, 8, 7, 22, 3, -1, 6, 8, 17, -1, 25, 3, 68, 4, 8, 14, 3, 11, 171, 6, 19, 5, 11, 11, 16, 6, 23, 19, 6, -1, 55, 3, 23, 10, 4, 26, 58, 6, 12, 3, 13, 3, 62, 9, 9, 4, 19, 62, 105, 9, 7, 24, 7, -1, 3, 17, 16, 12, 66, 21, 66, -1, 63, 65, 6, 28, 20, 20, 54, -1, 13, 92, 19, 21, 7, 9, 34, 36, 67, 5, 20, 5, 29, 62, 39, 6, 105

Offset

1,3

Comments

This sequence is similar to A073524, approximate squaring. However, for the cubing map it is easy to show that fractions of the form odd/2 never yield an integer. Hence if an iterate ever has this form, then we know it will never yield an integer. The computations, similar to A073524, must be done modulo n^max for some max > 2*a(n)+2.

Links

Table of n, a(n) for n=1..97.

Example

a(3)=3 because 4/3 -> 8/3 -> 64/3 -> 9728.

Mathematica

Stuck[x_] := OddQ[Numerator[x]] && (Denominator[x]==2); Table[lim=50; While[k=0; x=1+1/n; m=n^lim; While[2k<lim-3 && !IntegerQ[x] && !Stuck[x], x=Mod[x*Ceiling[x^2], m]; k++ ]; 2k>=lim-3, lim=2*lim]; If[Stuck[x], -1, k], {n, 200}]

Crossrefs

Cf. A118021 (n for which a(n)=-1).

Sequence in context: A232460 A337205 A287197 * A178619 A124572 A144880

Adjacent sequences: A118017 A118018 A118019 * A118021 A118022 A118023

Keyword

sign

Author

T. D. Noe, Apr 10 2006

Status

approved