[seqfan] Re: Non-square composite numbers

[Maurice.Clerc at WriteMe.com]

My informal description was not precise enough, sorry.
And the subject "Non-square composite numbers" is not very appropriate.

In fact I don't have just one sequence but a set of sequences, each one 
is built from the previous one.
Moreover I presented a sequence after having sorted it and it was not a 
good idea.

Here is the output of my algorithm
S(1) 2 3 6
  S(2) 2 3 6 12 18
  S(3) 2 3 6 12 18 24 36 54 72 108 216
  S(4) 2 3 6 12 18 24 36 54 72 108 216 48 144 432 162 324 648 1296 288 
864 2592 972 1944 3888 1728 5184 7776 5832 11664 15552 23328
  S(5) 2 3 6 12 18 24 36 54 72 108 216 48 144 432 162 324 648 1296 288 
864 2592 972 1944 3888 1728 5184 7776 5832 11664 15552 23328 *96* 576 
3456 10368 31104 46656 486 2916 17496 34992 69984 93312 139968 20736 
62208 186624 279936 104976 209952 419904 1152 6912 41472 124416 373248 
559872 839808 8748 52488 314928 629856 1259712 1119744 1679616 2519424 
3359232 5038848 13824 82944 248832 746496 2239488 6718464 10077696 
157464 944784 1889568 3779136 7558272 15116544 20155392 30233088 497664 
1492992 4478976 13436928 40310784 60466176 5668704 11337408 22674816 
45349632 90699264 8957952 26873856 80621568 120932352 181398528 68024448 
136048896 272097792 362797056

As you can see 96 does appear.

Now, let S3 be the infinite increasing sequence of 3-smooth numbers and 
{S3} the corresponding set.
As pointed out, for any k  {S(k)}-{2,3} is  a subset of  6*{S3}.

One could say we have here a funny way to generate 3-smooth numbers and, 
moreover, any element of S3 can be found  if k is big enough.
For k>5, as mentioned, S(k) does not seem to be in OEIS.

But, again, I am not sure it is of "general interest".




Le 21/07/2024 à 20:01, Allan Wechsler a écrit :
> Speaking only about Maurice's primary sequence: all entries after the 
> second are multiples of 6. If we look at these later elements divided 
> by 6, we get 1, 2, 3, 4, 6, 8, 9, 12 ... which certainly looks like 
> all the 3-smooth numbers. But then we skip 6*16 = 96, but I think this 
> is just a calculation error, because we certainly could have put in 96 
> after 72, because 96 = 2*48 and 2 and 48 are already in the sequence.
>
> So I conjecture that this sequence is just 2, 3, and then 6 times the 
> 3-smooth numbers. That's not to say this /doesn't/ belong in OEIS. The 
> definition is elegant enough that it wouldn't bother me if it got its 
> own entry.
>
> On Sun, Jul 21, 2024 at 12:34 PM Maurice.Clerc--- via SeqFan 
> <seqfan at list.seqfan.eu> wrote:
>
>     By studying how to pave the space with parallelepipeds I was led to
>     define the following sequence:
>
>     ----
>     List of integers so that each one after (2,3) is the product of k=2
>     different previous ones in the list. Duplicates are not allowed.
>     Example:
>     2 3 6 12 18 24 36 48 54 72 108 144 162 216 288 324 432 648 864 972
>     1296 1728
>
>     Note 1: they are not classical "composite numbers".
>     Note 2: it could be generalized for k>2.
>
>     ----
>     and also ;
>     - the similar one with duplicates.
>     - the similar ones without any square.
>
>     They don't seem to be in OEIS, but do you think it is worth adding one
>     of them (or more)?
>     I am not sure they are of "general interest", as required.
>
>     Regards
>     Maurice
>
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>