Issues with A069021

[Jona Christopher Sahnwaldt]

Hi everyone,

I think there are some issues with https://oeis.org/A069021.

Its current definition says: "Start of the first occurrence of n consecutive numbers divisible by a square greater than 1". The first few numbers in the sequence are 1, 8, 48, ...

Of course, 1 is not divisible by any number greater than 1, so either the definition is wrong, or 1 should not occur in the sequence. That's the first issue.

Until 2022, the definition was "Start of the first occurrence of n consecutive numbers divisible by a square". But that didn't work either: To make this definition compatible with 1 occuring in the list, 1 has to be divisible by a square, which means that we have to say 1^2 is a square. But if 1^2 is a square, then every number is divisible by a square, so any n consecutive numbers are all divisible by a square, and the sequence A069021 would have to be 1,1,1..., because the sequence 1,2 is a sequence of two consecutive numbers divisible by a square, and so on.

Gionata Neri raised this issue in 2017: "If 1 is in the sequence, then it is meant that 1^1 is a square, but in this case all the numbers would be divisible by a square?" https://oeis.org/history?seq=A069021

To fix the issue, we could apply the current definition "divisible by a square greater than 1" correctly and change the sequence from 1, 8, 48, ... to 4, 8, 48, ...

But that raises a second issue, because then A069021 is identical to A045882: "Smallest term of first run of (at least) n consecutive integers which are not squarefree." https://oeis.org/A045882

Cheers,

Christopher

[David desJardins]

If we are nitpicking, is the sequence (0) a run of 1 consecutive integers that are not squarefree? How about (-9)?

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[Sean A. Irvine]

Hi Christopher,

Given A045882, I don't think it is particularly worth worrying about.

However, our general rule is to preserve the data, so that would mean modifying the name to support a(1)=1. For example, we could write "a(1)=1, and for n>1, ..."

Sean.

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[Marc LeBrun]

I don't think just modifying the name in this way is sufficient to salvage A069021, because the value 4 is missing.

I would recommend inserting the 4 and also modifying the offset to start from n=0.  

That way it would at least become a consistent and accurate entry without too much more work.

After all, as the comment already in the "Crossrefs" notes, it's really just A045882 with an initial 1 prepended.

Granted, the concept of "zero consecutive numbers" is a bit of a stretch, but I also agree it isn't worth fussing overmuch with the wording (otherwise we'd also probably wind up changing "numbers" to "positive integers", and so on...)

--MLB

PS: Note that the convention of entering "lists" of values using the offset n=1 doesn't apply here, because the definition depends on the cardinality of n, not just its ordinality.

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