In Kansas, K-1 is a short connector from US 160/US 183 just south of Coldwater to the Oklahoma state line, where it becomes SH 34. It was originally much longer, but much of the mileage later became part of US 183.
In New Mexico, NM 1 was the original number from 1912 for what later became US 85. It has since been revived for a segment of former US 85 between Truth and Consequences and Soccoro that was bypassed by I-25. This allows NMDOT to avoid posting US 85 anywhere in New Mexico. The original NM 1 followed El Camino Real, which was the most important north-south overland route in territorial days, and may have been the longest state highway numbered in 1912. (I've never run across a document that states explicitly that routes were numbered from longest to shortest, but NM 2 and NM 3 were also very long.)
Texas has several Route 1s--NASA 1 and Ranch Road 1--but has not had a
SH 1 since 1952. Formally it is now a vest-pocket designation held by the Texas Transportation Commission and the TxDOT director, but it has effectively been retired. At its greatest extent, SH 1 ran diagonally from El Paso to Texarkana, so it was a worthy Route 1.