Achenes are not nuts.

(1) Achene. A small hard indehiscent fruit. The term is strictly only applied to those formed from one carpel, but is sometimes used for those formed from two carpels (e.g. the fruit of the Compositae). The latter is better termed a cypsela.

(2) Nut. This is similar to an achene, but is typically formed from two or three carpels (e.g. dock fruit).

www.sciencedirect.com/topics/…/achene

i. Achene - A one-seeded, dry, indehiscent fruit; the one seed is attached to the fruit wall at a single point.

ii. Nut - A dry, indehiscent, one seeded fruit similar to an achene but with the wall greatly thickened and hardened.

courses.botany.wisc.edu/…/LabWK03Fruitkey.html