[seqfan] A248034 and variants (Was: Digit-counters updating themselves)

[Antti Karttunen]

Dear SeqFans,

Taking Angelini's http://oeis.org/A248034 and Heinz's variant
http://oeis.org/A249009
as my starting points, I created some further variants of the same general
theme:

"Divide natural numbers to their constituent elements by some means, and
count the number of times such an element selected with some criteria from
a(n-1) occurs among the terms computed so far, up to and including a(n-1),
and let that count be the value of a(n)".

With A248034 the "elements" are digits 0-9 in base-10 representation of
natural numbers, and the element selected from a(n-1) is the least
significant digit. With A249009 it is otherwise same, but we count the
occurrences of the most significant digit of a(n-1).

For the latter, an obvious change is to use a factorial base, instead of
any fixed base, and we get:

A249069 a(n+1) gives the number of occurrences of the first digit of a(n)
in factorial base (i.e., A099563(a(n))) so far amongst the factorial base
representations of all the terms up to and including a(n), with a(0)=0.

another variant is to count the occurrences of maximal digit in factorial
expansion, as in:

A249070 a(n+1) gives the number of occurrences of the maximum digit of a(n)
in factorial base (i.e., A246359(a(n))) so far amongst the factorial base
representations of all the terms up to and including a(n), with a(0)=0.
(which has a slightly more interesting looking graph than the previous one).

Another possibility to avoid being limited to a finite number of elements
which to count is to use the runlengths of the binary expansion, as in:

A249144 a(0) = 0, after which a(n) gives the total number of runs of the
same length as the rightmost run in the binary representation of a(n-1)
[i.e., A136480(a(n-1))] among the binary expansions of all previous terms,
including the runs in a(n-1) itself.

and

A249146    a(0) = 0, after which a(n) gives the total number of runs of the
same length as the maximal run in the binary representation of a(n-1)
[i.e., A043276(a(n-1))] among the binary expansions of all previous terms,
including the runs in a(n-1) itself.

(Note the analogy with A249070).


Then of course, we can always count the prime factors, as in:

A249148    a(1) = 1, after which, if a(n-1) = 1, a(n) = 1 + the total
number of 1's that have occurred in the sequence so far, otherwise a(n) =
the total number of times the least prime dividing a(n-1) [i.e.,
A020639(a(n-1))] occurs as a divisor (counted with multiplicity for each
term) in the previous terms from a(1) up to and including a(n-1).

However, none of these have such a nice graph as Angelini's original
A248034, except very similar base-8 version A249068.


>From A249148, I proceeded to

A249336 a(1) = 1; for n>1, a(n) = number of values k in range 1 .. n-1 such
that {sum of prime indices in the prime factorization of a(k)} = {sum of
prime indices in the prime factorization of a(n-1)}, both counted with
multiplicity.

(and its minor variant A249337), where the theme is simpler now, to just
set a(n) as the count the number of terms a(k) from k = 1 to n-1 for which
f(a(k)) = f(a(n-1)), for some function f. In the above case that function
is A056239, which involves prime factorization.


But the graph is more interesting now:
http://oeis.org/A249336/graph
although I swear I have seen many similar ones already in OEIS, as the
theme certainly is old one.

Can somebody explain the different slopes of the "streamers" that cross
each other in that graph?
There seems to be roughly two different classes of them.


Best,

Antti



On Mon, Oct 20, 2014 at 4:39 AM, <seqfan-request at list.seqfan.eu> wrote:
>
>
> Message: 6
> Date: Sat, 18 Oct 2014 17:00:39 -0400
> From: Neil Sloane <njasloane at gmail.com>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Digit-counters updating themselves
> Message-ID:
>         <CAAOnSgSrRwqTNSZYs6PhYo_eCLPjTtDU=bx5oSX1piUYUHN=
iA at mail.gmail.com>
> Content-Type: text/plain; charset=UTF-8
>
> A week ago Eric created a lovely new sequence which Maximilian entered as
> A248034. It has a spectacular graph and it sounds pretty good too. I would
> like to be able to see more terms and listen to the rest of the music, if
> someone would create a b-file.
>
> I gave it the keywords look and hear.
>
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
> On Sat, Oct 11, 2014 at 3:28 PM, M. F. Hasler <oeis at hasler.fr> wrote:
>
> > Eric,
> >
> > my program seems to confirm your data (congrats !),
> > I submitted a proposal as https://oeis.org/draft/A248034
> >
> > -- Maximilian
> >
> > (PARI)
> >
> >
c=vector(10);print1(a=0);for(n=1,99,apply(d->c[d+1]++,if(a,digits(a)));print1(","a=c[1+a%10]))
> >
> > On Sat, Oct 11, 2014 at 1:59 PM, Eric Angelini <Eric.Angelini at kntv.be>
> > wrote:
> > >
> > >
> >
D=0,1,1,2,1,3,1,4,1,5,1,6,1,7,1,8,1,9,1,10,2,2,3,2,4,2,5,2,6,2,7,2,8,2,9,2,10,3,3,4,3,5,3,6,3,7,3,8,3,9,3,10,4,4,5,4,6,4,7,4,8,4,9,4,10,5,5,6,5,7,5,8,5,9,5,10,6,6,7,6,8,6,9,6,10,7,7,8,7,9,7,10,8,8,9,8,10,9,9,10,10,11,20,12,...
> > >
> > > Hello SeqFans,
> > > pick any comma in D.
> > > Immediately to the left of the comma
> > > there is a digit 'd'.
> > > Immediately to the right of the comma
> > > there is an integer d(n).
> > > D is such that there are d(n) digit 'd'
> > > so far in D [from the start of D up to the comma].
> > >
> > > In other words, the rightmost digit of d(n) is present d(n+1) times in
> > D, counting from d(1) to d(n).
> > >
> > > I'm wondering: do all integers appear
> > > at least once in D?
> > >
> > > P.-S.
> > > It is possible to compute similar sequences
> > > for every base. I guess the binary-one is:
> > >
> > > B = 0,1,1,10,10,11,110,100,110,111,1110,...
> > >
> > > Best,
> > > É.
> > >
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
>