[seqfan] Re: A248034 and variants (Was: Digit-counters updating themselves)
Eric Angelini
Eric.Angelini at kntv.be
Tue Oct 28 00:02:25 CET 2014
Don't worry, Antti, I didn't take
that for me -- I like noise!
;-)
Catapulté de mon aPhone
> Le 27 oct. 2014 à 23:58, "Antti Karttunen" <antti.karttunen at gmail.com> a écrit :
>
> Just a clarification: with "more noise" below I intended to be self-ironic,
> not to denigrate these kind of sequences in general. Also, as what comes to
> people who in general sneer on any base-related sequences, especially if
> using base ten (well, this set of people includes my former self): I
> specifically tried to demonstrate how the underlying idea is what counts,
> as it is often possible to apply it in many different contexts, some not
> "base" at all. Although involving something like prime factorization
> doesn't automatically make such sequences any better than good old decimal
> digit manipulation.
>
> Antti
>
>
> On Tue, Oct 28, 2014 at 12:40 AM, Antti Karttunen <antti.karttunen at gmail.com
>> wrote:
>
>>
>>
>> On Mon, Oct 27, 2014 at 10:57 PM, Antti Karttunen <
>> antti.karttunen at gmail.com> wrote:
>>
>>>
>>> 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.
>>
>> If I use A000010 (phi) as the function f in the above "sequence pattern"
>> (instead of A056239), and start with a(1) = 1, I will get the following
>> sequence (first 512 terms):
>>
>> 1, 1, 2, 3, 1, 4, 2, 5, 1, 6, 3, 4, 5, 2, 7, 1, 8, 3, 6, 7, 2, 9, 3, 8, 4,
>> 9, 4, 10, 5, 6, 11, 1, 10, 7, 5, 8, 9, 6, 12, 10, 11, 2, 11, 3, 13, 1, 12,
>> 12, 13, 2, 13, 3, 14, 7, 8, 14, 9, 10, 15, 1, 14, 11, 4, 15, 2, 15, 3, 16,
>> 4, 17, 1, 16, 5, 16, 6, 18, 12, 17, 2, 17, 3, 19, 1, 18, 13, 4, 20, 7, 14,
>> 15, 8, 18, 16, 9, 17, 4, 21, 5, 19, 2, 19, 3, 22, 5, 20, 10, 21, 6, 23, 1,
>> 20, 11, 6, 24, 12, 22, 7, 18, 19, 4, 25, 1, 21, 7, 20, 13, 8, 23, 2, 22, 8,
>> 24, 14, 21, 9, 22, 9, 23, 3, 26, 10, 25, 2, 23, 4, 27, 5, 26, 11, 10, 27,
>> 6, 28, 12, 28, 13, 14, 24, 15, 16, 17, 5, 29, 1, 24, 18, 25, 3, 29, 2, 25,
>> 4, 30, 19, 7, 26, 15, 20, 21, 16, 22, 11, 12, 30, 23, 5, 31, 1, 26, 17, 6,
>> 31, 2, 27, 8, 32, 7, 27, 9, 28, 18, 29, 3, 32, 8, 33, 5, 34, 9, 30, 24, 25,
>> 6, 33, 7, 31, 3, 34, 10, 35, 1, 28, 19, 10, 36, 20, 26, 21, 22, 13, 23, 6,
>> 35, 2, 29, 4, 36, 24, 27, 11, 14, 32, 11, 15, 28, 25, 8, 37, 1, 30, 29, 5,
>> 38, 12, 39, 3, 37, 2, 31, 4, 38, 13, 26, 27, 14, 33, 9, 34, 12, 40, 13, 28,
>> 29, 6, 39, 4, 40, 14, 35, 5, 41, 1, 32, 15, 30, 31, 5, 42, 30, 32, 16, 33,
>> 10, 43, 1, 33, 11, 16, 34, 17, 18, 36, 31, 6, 41, 2, 34, 19, 15, 35, 6, 42,
>> 32, 20, 36, 33, 12, 44, 13, 34, 21, 35, 7, 37, 3, 43, 2, 35, 8, 45, 9, 38,
>> 16, 37, 4, 44, 14, 39, 10, 46, 7, 40, 22, 17, 23, 8, 47, 1, 36, 36, 37, 5,
>> 48, 24, 38, 17, 25, 15, 39, 11, 18, 41, 3, 45, 12, 49, 3, 46, 9, 42, 38,
>> 18, 43, 4, 47, 2, 37, 6, 48, 26, 39, 13, 40, 27, 19, 20, 40, 28, 41, 4, 49,
>> 5, 50, 16, 41, 5, 51, 1, 38, 21, 42, 43, 6, 50, 17, 29, 7, 44, 18, 45, 14,
>> 46, 10, 52, 15, 42, 44, 19, 22, 19, 23, 11, 20, 43, 7, 47, 3, 51, 2, 39,
>> 16, 44, 20, 45, 17, 30, 46, 12, 53, 1, 40, 31, 7, 48, 32, 33, 21, 45, 18,
>> 49, 8, 54, 24, 47, 4, 52, 19, 25, 22, 21, 46, 13, 47, 5, 55, 6, 53, 2, 41,
>> 7, 50, 23, 14, 51, 3, 54, 26, 48, 34, 35, 20, 48, 36, 49, 9, 52, 21, 50,
>> 24, 49, 10, 56, 22, 22, 23, 15, 50, 25, 26, 51, 4, 55, 8, 57, 7, 53, 3, 56,
>> 23, 16, 51, 5
>>
>> which is not yet in OEIS. Should it be, or is just more of the same noise?
>> (Note how 1's can occur here also after nonprimes, like e.g. after 25).
>>
>> Antti
>>
>>
>>
>>>
>>> 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/
>
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