[seqfan] Re: Will this pattern continue?

[Hugo Pfoertner]

As expected, these are all rediscoveries of existing sequences, whereby a
trivial conversion is required in some cases. I will embed the respective
A-numbers in your post below. I leave it up to you to determine the
respective offsets.

On Sun, May 30, 2021 at 8:04 AM Ali Sada via SeqFan <seqfan at list.seqfan.eu>
wrote:

> Hi everyone,
>
> We take the twin primes and put them in pairs
> (3,5), (5,7), (11,13), (17,19), etc.
>
> We find the least triangular number that is a multiple of both primes in
> each pair
>
> 15, 105, 2145, 11628, 94395, 370230, 1565565, 3265290, 13263825, 16689753,
> 44674878, 62434725, 129757995, 168095280, 190173753, 334822503, 411256860,
> 659371455, 784892010, 1176876870, 1822721253
> (This is the best my computer and my computing skills can reach)
>
> Now, we divide each of these numbers by its two primes and we get
>
> 1, 3, 15, 36, 105, 210, 435, 630, 1275, 1431, 2346, 2775, 4005, 4560,
> 4851, 6441, 7140, 9045, 9870, 12090, 15051
>

This is 3 * A308344.


>
> So far, these are all triangular numbers. Will this pattern continue?
>
> These triangular numbers correspond to:
> 1, 2, 5, 8, 14, 20, 29, 35, 50, 53, 68, 74, 89, 95, 98, 113, 119, 134,
> 140, 155, 173 (I would like to add this sequence to the OEIS if the editors
> think it's suitable.)
>

3 * A002820 - 1


>
> If we exclude the first term, the differences between the terms are
> 3, 3, 6, 6, 9, 6, 15, 3, 15, 6, 15, 6, 3, 15, 6, 15, 6, 15, 18
>
> These are multiples of 3. Which is another potential pattern.
>

A145061 + 1


>
> We divide them by three and we get this "bareboned" sequence
> 1, 1, 2, 2, 3, 2, 5, 1, 5, 2, 5, 2, 1, 5, 2, 5, 2, 5, 6 ( I would love to
> add this sequence too if possible)
>

A160273

>
> I appreciate your feedback in advance.
>
> Best,
>
> Ali
>
>
>
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>