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We revealed 62 MAGs classified as key Arctic species, found only in the Arctic Ocean, showing the highest gene expression values and predicted to have habitat-specific traits. The Artic MAGs catalogue will inform our understanding of polar microorganisms that drive global biogeochemical cycles.This paper proves several novel formulas for the length of a Farey sequence of order n. The formulas use different trade-offs between iteration and recurrence and they range from simple to more complex. The paper also describes several iterative