How do you refer to those sequences $\{a_{n}\}_{n \in \mathbb{Z}^{+}}$ of integers that satisfy the condition $\text{gcd}(a_{m}, a_{n}) = a_{\text{gcd}(m,n)}$ for every $(m,n) \in \mathbb{Z}^{+} \times \mathbb{Z}^{+}$?

I believe some authors refer to 'em as "Mersenne sequences". Is this the most established term for integer sequences of that type?

Thanks in advance for your replies.

  • 10
    $\begingroup$ It appears that the standard term is "strong divisibility sequence". See here. $\endgroup$ Jun 14, 2022 at 19:15
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    $\begingroup$ Since the positive integers are a semigroup under gcd and such a sequence is a semigroup homomorphism a better name would be gcd-homomorphism. $\endgroup$ Jun 14, 2022 at 19:50
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    $\begingroup$ @BenjaminSteinberg You've come up with a better name, but since the terminology "strong divisibility sequence" is already well-embedded in the vast literature about divisibility sequences, it's probably best for the OP to use that terminology. And note for the OP, it is best if the title of your question is as specific as possible, so I'm going to edit it. If you don't like my edit, feel free to revert or change it to something better. $\endgroup$ Jun 14, 2022 at 20:33


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