# [seqfan] Re: Will this pattern continue?

Tue Jun 1 08:36:11 CEST 2021

```For the small sample I calculated, the pattern continued when I replaced one of the two primes with a perfect power of a prime. Is that a correct assumption?

For example, the smallest triangular number that is a multiple of 241 and 243 is 425167380. And when we divide it by the two initial numbers we get 7260, which is also a triangular number.

Best,
Ali
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On Sunday, May 30, 2021, 4:23:06 AM EDT, Frank Adams-watters via SeqFan <seqfan at list.seqfan.eu> wrote:

I'm a bit too tired to fully work this out right now, but:

Try looking at pairs of consecutive odd numbers, without regard to whether they are prime. You will find a much simpler relationship.

-----Original Message-----
From: Ali Sada via SeqFan <seqfan at list.seqfan.eu>
To: Sequence Fanatics Discussion List <seqfan at list.seqfan.eu>
Cc: Ali Sada <pemd70 at yahoo.com>
Sent: Sat, May 29, 2021 10:19 pm
Subject: [seqfan] Will this pattern continue?

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

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.)

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.

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)

Best,

Ali

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