Similarities between 2 divisor sequences

[Giorgos Kalogeropoulos]

A140747 and  A330757 look very similar...

They only differ for n = 60, 210, 264, 280, 300, 468, 495, 504, 585, 612, 616, 630, 660, 693, 728, 770, 780, 792, 816, 819, 880, 910, 912, 924, 990...

Do you think this should be a new sequence?

Best,

gk

[Daniel Mondot]

Very nice and intriguing find!

yes.

D.

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[Misha Lavrov]

It looks like the difference between these sequences comes from pairs of consecutive divisors that are not coprime, but whose ratio is not the same as the ratio of any previous pair of consecutive divisors. For example, in the case of 60, the pair (6,10) has a ratio of 3/5, which is counted by A330757; however, there's no pair with a ratio of 3/5 counted by A140747, because 3 and 5 are not consecutive divisors.

Going further, we first get two such pairs at 630, three pairs at 8580, four pairs at 10710, five pairs at 37620, and six pairs at 127512. Is the number of pairs (that is, the difference A330757(n) - A140747(n)) unbounded?

Best,

Misha

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