A390875 Positive integers k for which the 3-node triangle parity-routing process modulo k, starting from (1,1,1), does not halt for at least one of the 8 routing rules.

59, 107821, 3899141, 137724785

Offset

1,1

Comments

Three nodes sit on a triangle, each starting with value 1. At each step, the active node sends its value to its left or right neighbor based on the parity of its value and a per-node rule bit. The receiving node adds the sent value modulo k; if the result is 0, the process halts, otherwise the receiver becomes the new active node. With 3 nodes and 2 choices (left/right) per node, there are 2^3 = 8 possible rule sets. The sequence lists values of n for which at least one of the 8 rule sets causes the process to cycle forever instead of halting.

Verified exhaustively for k <= 137724785. No even k produces a cycle when starting values are (1,1,1) (empirical observation, unproven).

Links

Table of n, a(n) for n=1..4.

Reed Silverstein, Parity Lab: Interactive triangle simulation for target 59

Example

For k = 59, it cycles under rules {1,2,3,6} (0-indexed) with entry step 39 and cycle length 194.
For k = 107821, it cycles under rules {4,5} with entry step 156534 and cycle length 37866.
For k = 3899141, it cycles under rules {0,7} with entry step 2059355 and cycle length 1156485.
For k = 137724785, it cycles under rules {4,5} with entry step 296171191 and cycle length 85147918.

Crossrefs

Sequence in context: A258269 A094489 A235215 * A178066 A191947 A093403

Adjacent sequences: A390872 A390873 A390874 * A390876 A390877 A390878

Keyword

nonn,hard,more

Author

Reed Silverstein, Apr 12 2026

Status

approved