[seqfan] Re: Will this pattern continue?

Frank Adams-watters franktaw at netscape.net
Sun May 30 10:21:47 CEST 2021


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.

Franklin T. Adams-Watters


-----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)

I appreciate your feedback in advance. 

Best,

Ali 



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