[seqfan] Re: A025586/2

Thu Oct 17 20:18:10 CEST 2019

On 10/14/19, Hugo Pfoertner <yae9911 at gmail.com> wrote:
> There are many sequences divisible by a constant common factor, but
> typically only after discarding some initial terms. This also applies to
> A025586. But if discarding initial terms is an option, then why not discard
> two initial terms of A025586 and divide by 4? Or equivalently, A056959(n) /
> 4 ?

To abstract away from such a mundane details as with what constant the
terms of the sequence has been multiplied or not (or what constant has
been added to them), one can take the "Restricted Growth Sequence"
transform of the sequence (*), which will work as the
lexicographically earliest representative for all sequences that
partition natural numbers to exactly the same equivalence classes.
(*  provided it's not injective, because then we get just A000027
which is not very useful!)

E.g. in A025586's case we have terms, and we form A328314 as its
rgs-transform as follows:

n      A025586(n)    A328314(n)
1		1                1  (rgs-transforms always start with 1 as their first term)
2		2                2  (because 2 is a new term in A025586, and the
2nd distinct one)
3		16              3  (because 16 is a new term, and the 3rd distinct one)
4		4                4  (because 4 is a new term, and 4th distinct one)
5		16              3  (because 16 is NOT a new term, and occur for the
first time at n=3)
6		16              3  (ditto)
7		52              5  (because 52 is a new term, and the 5th distinct one)
8		8                6  (because 8 is a new term, and 6th distinct one)
9		52              5  (because 52 has occurred before, and is the 5th
distinct value in A025586)
10		16              3 (because 16 has occurred before, as the 3rd new
distinct one in A025586)
...
So we get the initial terms of A328314 as 1, 2, 3, 4, 3, 3, 5, 6, 5, 3, ...

Then, asking some help from my friendly djinn (giving him sequence
A328314 as input, from which he can in principle find all the
sequences that are its functions, but especially those that are
already in OEIS), I get the following list of sequences (plus some
noise, and some matches that might or might not be spurious, in any
case I manually removed them from this list).
A <=> B means that the djinn thinks that A can be computed from B and
vice versa, and A => B means that it thinks that B can be computed
from A, but not the other way around.

---

Searching matches for A328314 from seqs with at least 2 equivalence
classes and the second largest eq.class at least of size 2.
A328314 <=> (34,1,12) A328314 Lexicographically earliest infinite
sequence such that a(i) = a(j) => A025586(i) = A025586(j) for all i,
j.
A328314 <=> (23,5,7) A025586 Largest value in '3x+1' trajectory of n.
A328314 => (23,4,9) A222641 Number of iterations in Collatz (3x+1)
trajectory of n to reach 1 from the highest term.
A328314 => (22,5,7) A056959 In repeated iterations of function m->m/2
if m even, m->3m+1 if m odd, a(n) is maximum value achieved if
starting from n.
A328314 => (13,2,13) A095386 Largest prime factor of peak values of
3x+1 trajectory started at n.
A328314 => (7,72,4) A135282 Largest k such that 2^k appears in the
trajectory of the Collatz 3x+1 sequence started at n.
A328314 => (7,72,4) A226123 Number of terms of the form 2^k in
Collatz(3x+1) trajectory of n.
A328314 => (6,57,3) A232503 Largest power of 2 in the Collatz (3x+1)
trajectory of n.
A328314 => (3,73,14) A087976 a(n) = A001221(A025586(n)), the number of
distinct prime-factors of maximal term in 3x+1 iteration list started
at n.

----

And all those seem quite obvious cases, no surprises this time. It
will get more interesting when we combine values of different
functions to an ordered pair or tuple, elements of which the Djinn can
then use for its computations as it wishes.

Besides, many of the rgs-transforms make nice scatter plots as well,
like in this case:
https://oeis.org/A328314/graph
and one might wonder for this case where does that structure of
alternating heavy and sparsely dotted lines come from?

For some "looky" ones, please see:
https://oeis.org/search?q=%22Restricted+growth%22%7Crgs_transform+keyword%3Alook&sort=&language=&go=Search

Also, in the following case, by combining to an ordered pair some
classic number-theory related sequence (deficiency, A033879) and
classic base-2 related sequence (binary weight), in the resulting
scatter plot one can see the elements of both "domains" graphically
mixed, the rays emanating from the origin from the former, and a
square grid pattern from the latter:
https://oeis.org/A318310/graph

Maybe more elegant than above is:
https://oeis.org/A324344/graph
combining binary and primorial bases in its definition.

Yes, there are probably many sequences in OEIS that are restricted
growth sequence transforms of other sequences (including of
themselves, of course), but are not yet marked as such. Of course all
ordinal transforms satisfy also the restricted growth condition.

Best regards,

Antti

>
> On Mon, Oct 14, 2019 at 8:08 AM Frank Adams-watters via SeqFan <
> seqfan at list.seqfan.eu> wrote:
>
>> Yes (IMO). The half sequence should be added.
>>
>> Franklin T. Adams-Watters
>>
>>
>> -----Original Message-----
>> From: P. Michael Hutchins <pmh232 at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Sent: Sun, Oct 13, 2019 11:51 pm
>> Subject: [seqfan] A025586/2
>>
>> Every item in  A025586 <https://oeis.org/A025586> is even.  So we can
>> factor out the 2s.  My feeling is that such produces a "purer" sequence -
>> one that's more likely to be matched from some other domain.
>>
>> Is that enough to make a new sequence?
>>
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