The question was asked in Halacha about a person who misses lighting one of the nights of Chanukah? Does he continue to light on the subsequent nights, or is he out for the remainder of the holiday since he broke the continuity? It’s a funny question, but there’s actually an opinion brought down by one of the Rishonim, Rabeinu Shelomo who holds that in deed if for whatever reason you didn’t light and they didn’t light for you, you are then out. He holds it has to be like Sefirat HaOmer where you must have continuity. The Halacha however, we do not hold like Rabeinu Shelomo.
There’s a big difference between Sefirat HaOmer and the candles of Chanukah. By Sefirat HaOmer, the Torah says ‘Temimot’. The Torah says that it should be complete, and therefore if you miss a night in between so you are lacking the concept of ‘Temimot’, of the completeness of the Mitzvah. By the Candles of Chanukah however, each night is a separate Mitzvah on its own. The proof is that we say ‘SheAssa Nissim La’Avotenu BaYamim Hahem BaZeman HaZeh’ every single night. Which means that each night was a separate miracle. Therefore the Halacha, according to the Rama in siman 673:2 (written by Rabbi Moses ben Israel Isserles), if a person misses a night of lighting for whatever reason, he does pick up on the following night, not making up. You can’t make it up, but you just continue where you are up to.
Now there’s an interesting story told by one of the Rabbis. Before leaving the synagogue he used to ask the Shamosh (caretaker) what night of Chanukah are we up to? The Shamosh it seems was an Am Ha’Aretz (an un-intelligent person), and would always state what the previous night was. For example last night was 3. . So the Rabbi was asked why he had always asked the Shamosh. He answered that he wanted to be ‘Melamed Zechut’ (find a merit) for Am Yisrael (Jewish nation) by showing G-d how careful people were in the Mitzvot. The Rabbi said that the Shamosh figured that Ner (candles) Chanukah had an equal status of Sefirat Haomer, so therefore he would not mention which night it was because he thought it was a problem. We read the story and we think the Shamosh is not intelligent, but the Rabbi looks at it as a Zechut, that the people are so concerned that they want to do the Mitzvot the right way. The Shamosh was scared and each time said what the previous night was so that he wouldn’t run into a problem. So the Halacha, you can answer a person, if one asks what night are we up to. There’s no connection between Sefirat Haomer and Ner Chanukah.
