[seqfan] Re: SeqFan Digest, Vol 86, Issue 8
Dave Consiglio
davecons at gmail.com
Tue Nov 17 18:12:31 CET 2015
In response to Eric and the Pangram sequence:
I've calculated the sequence to 3 million terms (see attached code and text
file with 500K attached), and there is (so far) no repeating loop. There
are some long sections of repeating terms, though.
After 100,000 terms:
20, 20, 25, 5, 13, 25, 5, 25, 5, 4, 13, 25, 5, 25, 8, 5, 25, 8, 5, 13, 25,
8, 5, 25, 8, 5, 13, 13, 25, 8, 5, 25, 8, 5, 13, 25, 8, 5, 25, 8, 5, 13, 25,
8, 5, 25, 8, 5, 13, 25, 8, 5, 25, 8, 5, 13
After 200,000 terms:
3, 18, 27, 27, 7, 3, 18, 27, 27, 7, 3, 7, 27, 7, 3, 7, 27, 7, 3, 7, 27, 7,
3, 7, 27, 7, 9, 3, 27, 7, 3, 18, 27, 27, 7, 3, 27, 7
After 300,000 terms:
5, 6, 7, 29
etc.
Not entirely sure why there are these long patterns that eventually fail
(although my guess is that is has something to do with the floor function
sometimes moving numbers down a digit and sometimes not), but it's an
interesting sequence!
On Tue, Nov 17, 2015 at 12:10 PM, Dave Consiglio <davecons at gmail.com> wrote:
> In response to Eric and the Pangram sequence:
>
> I've calculated the sequence to 3 million terms (see attached code and
> text file), and there is (so far) no repeating loop. There are some long
> sections of repeating terms, though.
>
> After 100,000 terms:
>
> 20, 20, 25, 5, 13, 25, 5, 25, 5, 4, 13, 25, 5, 25, 8, 5, 25, 8, 5, 13, 25,
> 8, 5, 25, 8, 5, 13, 13, 25, 8, 5, 25, 8, 5, 13, 25, 8, 5, 25, 8, 5, 13, 25,
> 8, 5, 25, 8, 5, 13, 25, 8, 5, 25, 8, 5, 13
>
> After 200,000 terms:
>
> 3, 18, 27, 27, 7, 3, 18, 27, 27, 7, 3, 7, 27, 7, 3, 7, 27, 7, 3, 7, 27, 7,
> 3, 7, 27, 7, 9, 3, 27, 7, 3, 18, 27, 27, 7, 3, 27, 7
>
> After 300,000 terms:
>
> 5, 6, 7, 29
>
> etc.
>
> If you scroll to the end of the attached text file, it sure looks like
> there's a repeating pattern! But these things crop up all over the place.
> Not entirely sure why (although my guess is that is has something to do
> with the floor function sometimes moving numbers down a digit and sometimes
> not), but it's an interesting sequence!
>
>
>
>
>
>
> On Tue, Nov 17, 2015 at 10:11 AM, <seqfan-request at list.seqfan.eu> wrote:
>
>> Send SeqFan mailing list submissions to
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>>
>> Today's Topics:
>>
>> 1. 2,5,8,11,23, then what? (Neil Sloane)
>> 2. Re: A000108(n) ≡ 1 (mod 6) (William Keith)
>> 3. Re: 2,5,8,11,23, then what? (jean-paul allouche)
>> 4. Re: 2,5,8,11,23, then what? (Vladimir Shevelev)
>> 5. Re: A197680 (Vladimir Shevelev)
>> 6. Re: A197680 (Neil Sloane)
>> 7. Re: 2,5,8,11,23, then what? (jean-paul allouche)
>> 8. Re: Integral Apollonian circle packings (Rick Shepherd)
>> 9. Percentage of digits used so far (Eric Angelini)
>> 10. A081650 (Don Reble)
>> 11. A064513, A058201, and the graph degree/diameter problem.
>> (Allan Wechsler)
>> 12. Re: A081650 (israel at math.ubc.ca)
>> 13. Re: A081650 (Robert G. Wilson v)
>> 14. Re: A064513, A058201, and the graph degree/diameter problem.
>> (Chris Thompson)
>>
>>
>> ----------------------------------------------------------------------
>>
>> Message: 1
>> Date: Sun, 15 Nov 2015 11:44:06 -0500
>> From: Neil Sloane <njasloane at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] 2,5,8,11,23, then what?
>> Message-ID:
>> <CAAOnSgTj+cEf=
>> 8xQp2qr1PAjhtsoTLWXg3UVzJCh9WkPv84DaA at mail.gmail.com>
>> Content-Type: text/plain; charset=UTF-8
>>
>> Dear Seqfans,
>> In this paper on page 31 there is a sequence 2 5 8 11 23 ... which needs
>> more terms:
>>
>> Vladimir Shevelev, On the Basis Polynomials in the Theory of Permutations
>> with Prescribed Up-Down Structure, arXiv|math.CO/0801.0072.
>>
>> It appears that this is not yet in the OEIS. Maybe someone could extend
>> it?
>>
>> There is also an irregular triangle of coefficients in the Appendix, which
>> (correcting an obvious error) I have made into https://oeis.org/A263848.
>> This needs checking and could use more terms.
>>
>>
>> ------------------------------
>>
>> Message: 2
>> Date: Sun, 15 Nov 2015 11:54:42 -0500
>> From: William Keith <william.keith at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: A000108(n) ≡ 1 (mod 6)
>> Message-ID:
>> <
>> CA+G+0WX0+Bkxo-9pW55-kcis-Vkta_EkJOQJkiynDPHHLwKhVw at mail.gmail.com>
>> Content-Type: text/plain; charset=UTF-8
>>
>> Ah, right. I think Emmanuel has got the crux of it. I haven't filled out
>> the details but I think the carry rule solves the problem and proves
>> divisibility by 3 for m>8.
>>
>> If n = 2^m -1, then binomial(2n-2, n-1) = binomial(2^(m+1)-4, 2^m-1).
>> Observe that 2^m-1 is divisible by 3 iff m is even, and is divisible by
>> 3^1
>> strictly, unless m is 2 (mod 3), at which point the exponent of 3 is an
>> extended ruler sequence (2,2, 3, 2,2, 3, 2,2, 4, 2,2, 3, 2,2, 3, 2,2, 4,
>> 2,2, 3, 2,2, 3, 2,2, 5, etc.). The factor 1/(n+1) isn't pulling out any
>> powers of 3 since it's going to be 1/2^m, so I think divisibility by 3 is
>> guaranteed.
>>
>> Cordially,
>> William Keith
>>
>> On Sun, Nov 15, 2015 at 5:18 AM, Emmanuel Vantieghem <
>> emmanuelvantieghem at gmail.com> wrote:
>>
>> > Using Lucas' congruence for binomials I tested the conjecture up to m =
>> > 10000 (that is n = 2^10000 -1) and found it's true.
>> > By Lucas' congruence I mean that, modulo a prime number p, Binomial(a,b)
>> > === Product(Binomial(a(i),b(i)) where a(i) and b(i) are the 'digits'
>> of a
>> > resp. b in number base p). The test is very fast (it took less than a
>> > minute on my PC).
>> >
>> > 2015-11-15 7:35 GMT+01:00 L. Edson Jeffery <lejeffery2 at gmail.com>:
>> >
>> > > >I have tested the conjecture up to 250000 and found no other terms
>> > except
>> > > for 0, 1 & 31. Bob
>> > >
>> > >
>> > > Hello Bob,
>> > >
>> > > Are you saying that you tested the conjecture by checking each C(n)
>> > modulo
>> > > 6, for n=0..250000? If so, then there are only 17 odd C(n) within that
>> > > range (unless I miscounted), so the sample size is too small to draw
>> any
>> > > conclusions.
>> > >
>> > > We know that C(n) is odd if and only if n = 2^m - 1, where m >= 0 is
>> an
>> > > integer. Using that, we could test the conjecture only for those
>> C(2^m -
>> > > 1), m=0,1,... . But the number of digits of C(2^(m+1) - 1) appears to
>> be
>> > > roughly twice that of C(2^m - 1) which causes obvious problems with
>> > > computations. The number of digits of C(2^m - 1), for m >= 0 (as
>> counted
>> > by
>> > > Mathematica), is the sequence starting
>> > >
>> > > {1, 1, 1, 3, 7, 17, 35, 74, 150, 304, 612, 1228, 2460, 4926, 9857,
>> 19721,
>> > > 39449, 78905, 157818, 315644, 631296, 1262601, 2525212, 5050435,
>> > 10100879,
>> > > 20201769, 40403550, 80807112, ...}
>> > >
>> > > I tested the conjecture for C(2^m - 1) up to the still very small m =
>> 31,
>> > > or n = 2147483647, using Mathematica, but got an overflow error (or
>> > > something like that) because C(2147483647) is already so large: I
>> guess
>> > > that C(2^31 - 1) has a little more than 1292913800 digits. So is
>> there a
>> > > better way to test these numbers?
>> > >
>> > > Finally, note that if C(n) is odd, then C(n) == 1 (mod 3) implies that
>> > C(n)
>> > > == 1 (mod 6) (or 1 (mod 12)) because also C(n) == 1 (mod 4).
>> > >
>> > > Ed Jeffery
>> > >
>> > > _______________________________________________
>> > >
>> > > Seqfan Mailing list - http://list.seqfan.eu/
>> > >
>> >
>> > _______________________________________________
>> >
>> > Seqfan Mailing list - http://list.seqfan.eu/
>> >
>>
>>
>> ------------------------------
>>
>> Message: 3
>> Date: Sun, 15 Nov 2015 19:44:06 +0100
>> From: jean-paul allouche <jean-paul.allouche at imj-prg.fr>
>> To: <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: 2,5,8,11,23, then what?
>> Message-ID: <5648D276.2040800 at imj-prg.fr>
>> Content-Type: text/plain; charset="utf-8"; format=flowed
>>
>> Dear all
>>
>> It might be worth adding that this sequence appears
>> in the published paper:
>> http://www.emis.de/journals/INTEGERS/papers/m1/m1.pdf
>>
>> best
>> jp
>>
>> Le 15/11/15 17:44, Neil Sloane a écrit :
>> > Dear Seqfans,
>> > In this paper on page 31 there is a sequence 2 5 8 11 23 ... which needs
>> > more terms:
>> >
>> > Vladimir Shevelev, On the Basis Polynomials in the Theory of
>> Permutations
>> > with Prescribed Up-Down Structure, arXiv|math.CO/0801.0072.
>> >
>> > It appears that this is not yet in the OEIS. Maybe someone could extend
>> it?
>> >
>> > There is also an irregular triangle of coefficients in the Appendix,
>> which
>> > (correcting an obvious error) I have made into https://oeis.org/A263848
>> .
>> > This needs checking and could use more terms.
>> >
>> > _______________________________________________
>> >
>> > Seqfan Mailing list - http://list.seqfan.eu/
>>
>>
>>
>> ------------------------------
>>
>> Message: 4
>> Date: Sun, 15 Nov 2015 19:06:50 +0000
>> From: Vladimir Shevelev <shevelev at bgu.ac.il>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: 2,5,8,11,23, then what?
>> Message-ID:
>> <7416D8E96580CA458C090F5FB86540311C8ADE67 at hawk3.auth.ad.bgu.ac.il
>> >
>> Content-Type: text/plain; charset="utf-8"
>>
>> Thank you, Jean-Paul;
>> here this sequence is in Section 11, point 3, p.33.
>> Besides, many additional facts on this topic
>> one can find in V. Shevelev and J. Spilker, Up-down
>> coefficients for permutations. Elemente der Mathematik,
>> Vol. 68 (2013), no.3, 115–127 (I have no a link).
>>
>> Best regards,
>> Vladimir
>>
>> ________________________________________
>> From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of jean-paul
>> allouche [jean-paul.allouche at imj-prg.fr]
>> Sent: 15 November 2015 20:44
>> To: seqfan at list.seqfan.eu
>> Subject: [seqfan] Re: 2,5,8,11,23, then what?
>>
>> Dear all
>>
>> It might be worth adding that this sequence appears
>> in the published paper:
>> http://www.emis.de/journals/INTEGERS/papers/m1/m1.pdf
>>
>> best
>> jp
>>
>> Le 15/11/15 17:44, Neil Sloane a écrit :
>> > Dear Seqfans,
>> > In this paper on page 31 there is a sequence 2 5 8 11 23 ... which needs
>> > more terms:
>> >
>> > Vladimir Shevelev, On the Basis Polynomials in the Theory of
>> Permutations
>> > with Prescribed Up-Down Structure, arXiv|math.CO/0801.0072.
>> >
>> > It appears that this is not yet in the OEIS. Maybe someone could extend
>> it?
>> >
>> > There is also an irregular triangle of coefficients in the Appendix,
>> which
>> > (correcting an obvious error) I have made into https://oeis.org/A263848
>> .
>> > This needs checking and could use more terms.
>> >
>> > _______________________________________________
>> >
>> > Seqfan Mailing list - http://list.seqfan.eu/
>>
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>> ------------------------------
>>
>> Message: 5
>> Date: Sun, 15 Nov 2015 19:11:32 +0000
>> From: Vladimir Shevelev <shevelev at bgu.ac.il>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: A197680
>> Message-ID:
>> <7416D8E96580CA458C090F5FB86540311C8ADE7C at hawk3.auth.ad.bgu.ac.il
>> >
>> Content-Type: text/plain; charset="iso-8859-8-i"
>>
>> Neil, thank you!
>> Of course, you meant that 256 is not a member
>> (it is in alternative 2).
>>
>> Best regards,
>> Vladimir
>> ________________________________________
>> From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Neil Sloane [
>> njasloane at gmail.com]
>> Sent: 15 November 2015 17:30
>> To: Sequence Fanatics Discussion list
>> Subject: [seqfan] Re: A197680
>>
>> Alternative 1 is correct. I don't think there is any ambiguity in the
>> name.
>>
>> 256 is a member
>>
>> I added a comment to explain the definition
>>
>> 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 Sun, Nov 15, 2015 at 3:50 AM, Vladimir Shevelev <shevelev at bgu.ac.il>
>> wrote:
>>
>> > Is 256 a term of A197680?
>> > It seems that the name "Numbers whose
>> > prime factors have powers that are squares"
>> > is a bit ambiguous. There are two variants
>> > of the understanding it.
>> > 1) All exponents in prime power factorization
>> > of terms are squares;
>> > 2) All such exponents are squares or of the form
>> > k*m^2, where k,m>=2 ("have squares").
>> > What is right?
>> > P.S. By 1), in Oct 14 I suggested a reformulation
>> > of the name: "Exponentially squares", but yet my
>> > comments have not been published.
>> >
>> > Best regards,
>> > Vladimir
>> >
>> > _______________________________________________
>> >
>> > Seqfan Mailing list - http://list.seqfan.eu/
>> >
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>>
>> ------------------------------
>>
>> Message: 6
>> Date: Sun, 15 Nov 2015 14:53:05 -0500
>> From: Neil Sloane <njasloane at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: A197680
>> Message-ID:
>> <
>> CAAOnSgSy05hVUotuY2xL57gQztNJJ052FdrNeUqqVOkYr0snOQ at mail.gmail.com>
>> Content-Type: text/plain; charset=UTF-8
>>
>> yes, sorry, my mistake
>>
>> 256 = 2^8 and 8 is not a square
>>
>> 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 Sun, Nov 15, 2015 at 2:11 PM, Vladimir Shevelev <shevelev at bgu.ac.il>
>> wrote:
>>
>> > Neil, thank you!
>> > Of course, you meant that 256 is not a member
>> > (it is in alternative 2).
>> >
>> > Best regards,
>> > Vladimir
>> > ________________________________________
>> > From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Neil Sloane [
>> > njasloane at gmail.com]
>> > Sent: 15 November 2015 17:30
>> > To: Sequence Fanatics Discussion list
>> > Subject: [seqfan] Re: A197680
>> >
>> > Alternative 1 is correct. I don't think there is any ambiguity in the
>> > name.
>> >
>> > 256 is a member
>> >
>> > I added a comment to explain the definition
>> >
>> > 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 Sun, Nov 15, 2015 at 3:50 AM, Vladimir Shevelev <shevelev at bgu.ac.il>
>> > wrote:
>> >
>> > > Is 256 a term of A197680?
>> > > It seems that the name "Numbers whose
>> > > prime factors have powers that are squares"
>> > > is a bit ambiguous. There are two variants
>> > > of the understanding it.
>> > > 1) All exponents in prime power factorization
>> > > of terms are squares;
>> > > 2) All such exponents are squares or of the form
>> > > k*m^2, where k,m>=2 ("have squares").
>> > > What is right?
>> > > P.S. By 1), in Oct 14 I suggested a reformulation
>> > > of the name: "Exponentially squares", but yet my
>> > > comments have not been published.
>> > >
>> > > Best regards,
>> > > Vladimir
>> > >
>> > > _______________________________________________
>> > >
>> > > Seqfan Mailing list - http://list.seqfan.eu/
>> > >
>> >
>> > _______________________________________________
>> >
>> > Seqfan Mailing list - http://list.seqfan.eu/
>> >
>> > _______________________________________________
>> >
>> > Seqfan Mailing list - http://list.seqfan.eu/
>> >
>>
>>
>> ------------------------------
>>
>> Message: 7
>> Date: Sun, 15 Nov 2015 21:09:25 +0100
>> From: jean-paul allouche <jean-paul.allouche at imj-prg.fr>
>> To: <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: 2,5,8,11,23, then what?
>> Message-ID: <5648E675.6020408 at imj-prg.fr>
>> Content-Type: text/plain; charset="utf-8"; format=flowed
>>
>> Hi Vladimir
>> The link for this other paper of yours (for subscribers only) is
>>
>> http://www.ems-ph.org/journals/show_abstract.php?issn=0013-6018&vol=68&iss=3&rank=4
>>
>> best
>> jp
>>
>> Le 15/11/15 20:06, Vladimir Shevelev a écrit :
>> > Thank you, Jean-Paul;
>> > here this sequence is in Section 11, point 3, p.33.
>> > Besides, many additional facts on this topic
>> > one can find in V. Shevelev and J. Spilker, Up-down
>> > coefficients for permutations. Elemente der Mathematik,
>> > Vol. 68 (2013), no.3, 115–127 (I have no a link).
>> >
>> > Best regards,
>> > Vladimir
>> >
>> > ________________________________________
>> > From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of jean-paul
>> allouche [jean-paul.allouche at imj-prg.fr]
>> > Sent: 15 November 2015 20:44
>> > To: seqfan at list.seqfan.eu
>> > Subject: [seqfan] Re: 2,5,8,11,23, then what?
>> >
>> > Dear all
>> >
>> > It might be worth adding that this sequence appears
>> > in the published paper:
>> > http://www.emis.de/journals/INTEGERS/papers/m1/m1.pdf
>> >
>> > best
>> > jp
>> >
>> > Le 15/11/15 17:44, Neil Sloane a écrit :
>> >> Dear Seqfans,
>> >> In this paper on page 31 there is a sequence 2 5 8 11 23 ... which
>> needs
>> >> more terms:
>> >>
>> >> Vladimir Shevelev, On the Basis Polynomials in the Theory of
>> Permutations
>> >> with Prescribed Up-Down Structure, arXiv|math.CO/0801.0072.
>> >>
>> >> It appears that this is not yet in the OEIS. Maybe someone could
>> extend it?
>> >>
>> >> There is also an irregular triangle of coefficients in the Appendix,
>> which
>> >> (correcting an obvious error) I have made into
>> https://oeis.org/A263848.
>> >> This needs checking and could use more terms.
>> >>
>> >> _______________________________________________
>> >>
>> >> Seqfan Mailing list - http://list.seqfan.eu/
>> >
>> > _______________________________________________
>> >
>> > Seqfan Mailing list - http://list.seqfan.eu/
>> >
>> > _______________________________________________
>> >
>> > Seqfan Mailing list - http://list.seqfan.eu/
>>
>>
>>
>> ------------------------------
>>
>> Message: 8
>> Date: Sun, 15 Nov 2015 16:17:17 -0500
>> From: Rick Shepherd <rlshepherd2 at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: Integral Apollonian circle packings
>> Message-ID:
>> <CAAaTU+g+FHwpzrL+i60ocBT4u1M45a6CHoBt=
>> OXhTzZr-o8+nQ at mail.gmail.com>
>> Content-Type: text/plain; charset=UTF-8
>>
>> Thanks to all for the references and the answer!
>>
>> Although my reading about and understanding of this topic is still
>> limited,
>> I believe it may yet be of some interest to explore with sequences where
>> (i.e., at what generation) these duplicate curvatures appear. (I don't
>> have time to do this now.)
>>
>> An easy observation:
>> By the definitions of integral Apollonian circle packing (or Apollonian
>> gasket) and of curvature (or bend), then each such packing of an outer
>> circle with curvature -n gives an infinite sum of reciprocals of squares
>> totaling 1/n^2. For example, 1 = sum_{n=2..infty} A042946(n)/A042944(n)^2
>> (assuming indices matched as they should but don't currently) as these
>> sequences describe a unit circle packing. Since Apollonian circle
>> packings
>> could be nested (to arbitrary depths), an infinite number of such infinite
>> sums could potentially be derived for each total -- while ultimately also
>> making the largest (duplicate) circles in this type of packing as small as
>> desired.
>>
>> Rick
>>
>>
>> ------------------------------
>>
>> Message: 9
>> Date: Tue, 17 Nov 2015 00:38:10 +0100
>> From: Eric Angelini <Eric.Angelini at kntv.be>
>> To: Sequence Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Percentage of digits used so far
>> Message-ID: <E00372D0-5B2D-4D08-AA78-F1EABDF09311 at kntv.be>
>> Content-Type: text/plain; charset="utf-8"
>>
>>
>> Hello SeqFans,
>> a recent post by Lee Sallows (link at
>> the end of this msg) gave me the idea of P;
>>
>> P=100,50,66,25,60,16,28,12,22,30,36,
>> 16,46,14,6,6,17,22,26,5,23,9,21,16,24,
>> 3,25,17,6,23,29,18,3,...
>>
>> The p(n)th integer of P is the fraction A/B (in percents, decimals
>> erased) with:
>> A = the number of times we have
>> used the (n)th digit of P so far;
>> B = the overall quantity of digits used
>> so far to write P.
>>
>> Example:
>> 100 means that we have used one "1"
>> to write 1 digit so far --> 1/1 is 100%
>> 50 means that we have used one "0"
>> to write 2 digits so far --> 1/2 is 50%
>> 66 means that we have used two "0"
>> to write 3 digits so far --> 2/3 is 66%
>> 25 means that we have used one "5"
>> to write 4 digits so far --> 1/4 is 25%
>> 60 means that we have used three "0"
>> to write 5 digits so far --> 3/5 is 60%
>> 16 means that we have used one "6"
>> to write 6 digits so far --> 1/6 is 16%
>> 28 means that we have used two "6"
>> to write 7 digits so far --> 2/7 is 28%
>> 12 means that we have used one "2"
>> to write 8 digits so far --> 1/8 is 12%
>> 22 means that we have used two "5"
>> to write 9 digits so far --> 2/9 is 22%
>> 30 means that we have used three "6"
>> to write 10 digits so far --> 3/10 is 30%
>> ...
>>
>> Does P enter into some kind of loop
>> at some point?
>> Best,
>> É.
>>
>> Lee's link:
>> http://www.futilitycloset.com/2015/11/16/a-new-pangram/
>>
>> Catapulté de mon aPhone
>>
>> ------------------------------
>>
>> Message: 10
>> Date: Mon, 16 Nov 2015 15:56:50 -0700
>> From: Don Reble <djr at nk.ca>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] A081650
>> Message-ID: <564A5F32.7060905 at nk.ca>
>> Content-Type: text/plain; charset=UTF-8; format=flowed
>>
>> Seqfans:
>>
>> > %S A081650 2,5,13,73,409,801,1584,2241,30601,30601,78409,156825, ...
>> > %N A081650 Least nonsquare congruent to i^2 (mod k^2) for all 0<k<=n,
>> > i being any integer.
>> > %O A081650 1,1
>>
>> Why is a(4)=73, instead of 52?
>>
>> --
>> don r djr at .nk.ca
>>
>>
>> ------------------------------
>>
>> Message: 11
>> Date: Mon, 16 Nov 2015 13:53:28 -0500
>> From: Allan Wechsler <acwacw at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] A064513, A058201, and the graph degree/diameter
>> problem.
>> Message-ID:
>> <
>> CADy-sGFqG9z_+keC9sJ-hrp_vOxxRRi45H-pOQ6B93jngMFY9Q at mail.gmail.com>
>> Content-Type: text/plain; charset=UTF-8
>>
>> The degree/diameter problem is a classic in extremal graph theory. For a
>> given degree d and diameter k, we are asked, what is the largest number of
>> nodes that a graph can have?
>>
>> In particular, even when the required diameter k=2, we don't know the
>> answer to this question beyond d=7 or so.
>>
>> OEIS has two sequences that purport to give the data we have; these are
>> A058201 and A064513. They give values for d between 1 and 5, inclusive,
>> and
>> they agree that a(1)=2, a(2)=4, a(3)=10, and a(4) = 15. But they disagree
>> on a(5): the older sequence says it's 24; the newer one says it's 25.
>>
>> The first question this raises for me is, why do they say a(2) = 4?
>> Doesn't
>> a cyclic graph of order 5 have degree 2 and diameter 2?
>>
>> Wikipedia has articles on the degree/diameter problem. The first column of
>> their table (
>>
>> https://en.wikipedia.org/wiki/Table_of_the_largest_known_graphs_of_a_given_diameter_and_maximal_degree
>> )
>> gives values 2, 5, 10, 15, 25, 32, 50. So their a(5) agrees with the later
>> sequence on OEIS, but their a(2) (and *my* a(2), for that matter!)
>> disagrees with *both* versions on OEIS.
>>
>> I propose that we mark A058201 as wrong, but leave it as it stands,
>> otherwise, and fix A064513 to set a(2)=5 and fill in a(6) and a(7) from
>> the
>> sources referenced in the Wikipedia articles.
>>
>>
>> ------------------------------
>>
>> Message: 12
>> Date: 16 Nov 2015 23:04:02 -0800
>> From: israel at math.ubc.ca
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: A081650
>> Message-ID: <Prayer.1.3.5.1511162304020.28044 at viol.math.ubc.ca>
>> Content-Type: text/plain; format=flowed; charset=ISO-8859-1
>>
>> >From the program, it seems to be that "(mod k^2)" is being used as a
>> function rather than a relation, so even though 52 is congruent to a
>> square
>> mod 9 (i.e. 52 == 4^2 mod 9), 52 mod 9 = 7 is not a square.
>>
>> Since this is an old sequence, I don't think its Data should be changed,
>> rather the definition should be clarified. The sequence 2, 5, 13, 52, 241,
>> 241, 436, 1009, 1009, 1009, ... corresponding to congruence as a relation
>> does not seem to be in the OEIS: perhaps you'd like to contribute it.
>>
>> Cheers,
>> Robert
>>
>> On Nov 16 2015, Don Reble wrote:
>>
>> >Seqfans:
>> >
>> > > %S A081650 2,5,13,73,409,801,1584,2241,30601,30601,78409,156825, ...
>> > > %N A081650 Least nonsquare congruent to i^2 (mod k^2) for all 0<k<=n,
>> > > i being any integer.
>> > > %O A081650 1,1
>> >
>> > Why is a(4)=73, instead of 52?
>> >
>> >
>>
>>
>> ------------------------------
>>
>> Message: 13
>> Date: Tue, 17 Nov 2015 10:03:23 -0500
>> From: "Robert G. Wilson v" <rgwv at rgwv.com>
>> To: "'Sequence Fanatics Discussion list'" <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: A081650
>> Message-ID: <002a01d12149$1aee0230$50ca0690$@com>
>> Content-Type: text/plain; charset="utf-8"
>>
>> Instead of congruent, how about remainder?
>>
>> -----Original Message-----
>> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of
>> israel at math.ubc.ca
>> Sent: Tuesday, November 17, 2015 2:04 AM
>> To: Sequence Fanatics Discussion list
>> Subject: [seqfan] Re: A081650
>>
>> >From the program, it seems to be that "(mod k^2)" is being used as a
>> function rather than a relation, so even though 52 is congruent to a square
>> mod 9 (i.e. 52 == 4^2 mod 9), 52 mod 9 = 7 is not a square.
>>
>> Since this is an old sequence, I don't think its Data should be changed,
>> rather the definition should be clarified. The sequence 2, 5, 13, 52, 241,
>> 241, 436, 1009, 1009, 1009, ... corresponding to congruence as a relation
>> does not seem to be in the OEIS: perhaps you'd like to contribute it.
>>
>> Cheers,
>> Robert
>>
>> On Nov 16 2015, Don Reble wrote:
>>
>> >Seqfans:
>> >
>> > > %S A081650 2,5,13,73,409,801,1584,2241,30601,30601,78409,156825, ...
>> > > %N A081650 Least nonsquare congruent to i^2 (mod k^2) for all 0<k<=n,
>> > > i being any integer.
>> > > %O A081650 1,1
>> >
>> > Why is a(4)=73, instead of 52?
>> >
>> >
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>>
>>
>> ------------------------------
>>
>> Message: 14
>> Date: 17 Nov 2015 14:37:12 +0000
>> From: Chris Thompson <cet1 at cam.ac.uk>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: [seqfan] Re: A064513, A058201, and the graph degree/diameter
>> problem.
>> Message-ID: <Prayer.1.3.5.1511171437120.29531 at hermes-2.csi.cam.ac.uk>
>> Content-Type: text/plain; format=flowed; charset=ISO-8859-1
>>
>> On Nov 17 2015, Allan Wechsler wrote:
>>
>> [... snipped...]
>> >Wikipedia has articles on the degree/diameter problem. The first column
>> of
>> >their table (
>>
>> https://en.wikipedia.org/wiki/Table_of_the_largest_known_graphs_of_a_given_diameter_and_maximal_degree
>> >) gives values 2, 5, 10, 15, 25, 32, 50. So their a(5) agrees with the
>> later
>> >sequence on OEIS, but their a(2) (and *my* a(2), for that matter!)
>> >disagrees with *both* versions on OEIS.
>>
>> I read the Wikipedia page as saying that a(5) = 24 - admittedly the dark
>> blue background could easily lead to misreading!
>>
>> In any case, that agrees with the EJC Dynamic Survey (updated 2013). The
>> Wikipedia link to that seems to be broken: a working one is
>>
>> http://www.combinatorics.org/ojs/index.php/eljc/article/view/DS14
>>
>> This also confirms a(6) = 32 as optimal.
>>
>> --
>> Chris Thompson
>> Email: cet1 at cam.ac.uk
>>
>>
>> ------------------------------
>>
>> Subject: Digest Footer
>>
>> _______________________________________________
>> SeqFan mailing list
>> SeqFan at list.seqfan.eu
>> http://list.seqfan.eu/cgi-bin/mailman/listinfo/seqfan
>>
>> ------------------------------
>>
>> End of SeqFan Digest, Vol 86, Issue 8
>> *************************************
>>
>
>
-------------- next part --------------
100, 50, 66, 25, 60, 16, 28, 12, 22, 30, 36, 16, 30, 14, 6, 18, 17, 22, 26, 5, 23, 9, 21, 16, 24, 11, 22, 17, 3, 23, 19, 6, 21, 2, 17, 19, 21, 21, 7, 22, 9, 2, 23, 18, 20, 19, 23, 4, 20, 22, 23, 25, 22, 3, 9, 25, 10, 22, 3, 16, 24, 22, 25, 23, 4, 24, 4, 25, 24, 25, 25, 5, 26, 27, 5, 27, 28, 8, 24, 3, 28, 8, 24, 5, 28, 9, 3, 28, 8, 28, 29, 30, 9, 30, 4, 31, 31, 10, 6, 32, 4, 20, 8, 31, 32, 10, 20, 10, 32, 3, 32, 33, 33, 5, 33, 10, 4, 33, 5, 5, 33, 5, 34, 6, 34, 6, 34, 7, 7, 34, 9, 34, 3, 8, 34, 4, 35, 2, 3, 35, 6, 9, 34, 4, 4, 34, 6, 8, 34, 5, 4, 9, 34, 5, 6, 34, 7, 34, 5, 9, 6, 5, 9, 6, 6, 10, 13, 10, 14, 14, 7, 7, 10, 32, 6, 32, 7, 6, 11, 14, 11, 31, 14, 7, 31, 8, 14, 8, 11, 31, 12, 12, 31, 12, 13, 13, 14, 6, 14, 15, 14, 8, 6, 15, 15, 6, 7, 15, 16, 7, 16, 6, 6, 16, 6, 6, 17, 7, 3, 3, 17, 7, 4, 17, 8, 17, 5, 17, 8, 8, 18, 7, 26, 18, 18, 7, 6, 4, 18, 8, 9, 9, 18, 9, 6, 5, 19, 10, 7, 10, 4, 19, 10, 7, 7, 19, 10, 3, 19, 11, 8, 4, 7, 8, 5, 7, 7, 11, 6, 11, 18, 11, 6, 12, 10, 12, 11, 3, 3, 12, 7, 18, 22, 7, 18, 22, 4, 7, 12, 12, 13, 10, 13, 13, 18, 13, 14, 11, 4, 18, 14, 4, 14, 11, 5, 14, 14, 18, 15, 15, 20, 15, 21, 18, 15, 16, 21, 16, 18, 16, 18, 16, 10, 7, 16, 11, 17, 7, 17, 11, 5, 7, 17, 7, 17, 7, 7, 4, 17, 7, 17, 7, 4, 18, 8, 8, 8, 18, 8, 9, 9, 18, 4, 4, 16, 17, 18, 5, 5, 10, 18, 5, 4, 18, 5, 7, 18, 5, 5, 5, 18, 5, 6, 18, 9, 18, 5, 18, 6, 6, 9, 10, 18, 6, 6, 3, 4, 18, 6, 4, 9, 7, 18, 4, 18, 5, 6, 19, 5, 10, 19, 4, 19, 5, 6, 6, 19, 4, 19, 5, 15, 19, 5, 19, 20, 6, 9, 6, 6, 6, 6, 7, 19, 20, 8, 20, 20, 20, 6, 20, 20, 8, 21, 15, 21, 5, 21, 15, 21, 21, 14, 14, 21, 15, 6, 21, 6, 16, 16, 7, 21, 6, 16, 16, 9, 7, 21, 16, 21, 16, 21, 14, 22, 5, 22, 14, 22, 14, 22, 6, 22, 14, 22, 9, 22, 22, 9, 23, 6, 23, 9, 9, 23, 9, 23, 23, 6, 23, 9, 23, 9, 23, 6, 23, 6, 23, 6, 15, 5, 24, 6, 15, 24, 24, 6, 24, 6, 24, 7, 15, 24, 24, 7, 24, 6, 24, 7, 24, 7, 25, 8, 25, 5, 6, 25, 8, 25, 25, 25, 6, 6, 25, 6, 25, 25, 6, 7, 25, 7, 7, 25, 7, 7, 7, 9, 25, 8, 8, 25, 8, 8, 9, 25, 6, 7, 7, 7, 25, 7, 7, 3, 4, 25, 7, 9, 9, 25, 7, 25, 8, 25, 7, 6, 6, 25, 5, 25, 7, 6, 9, 25, 8, 6, 8, 25, 8, 6, 6, 7, 25, 8, 7, 7, 25, 8, 4, 25, 8, 7, 26, 8, 7, 7, 4, 25, 4, 26, 8, 7, 8, 11, 8, 25, 8, 8, 9, 4, 8, 25, 8, 9, 25, 8, 7, 8, 25, 4, 7, 25, 4, 25, 4, 9, 26, 4, 7, 8, 8, 25, 4, 9, 26, 4, 7, 26, 7, 26, 4, 7, 26, 5, 12, 4, 8, 5, 8, 8, 8, 8, 7, 25, 5, 12, 4, 8, 12, 5, 12, 5, 12, 5, 8, 12, 5, 12, 5, 8, 12, 25, 25, 7, 13, 25, 7, 13, 25, 25, 7, 13, 25, 13, 25, 25, 8, 25, 8, 13, 25, 26, 7, 8, 13, 26, 8, 26, 8, 26, 8, 7, 13, 26, 8, 26, 9, 26, 9, 5, 7, 13, 26, 26, 9, 13, 26, 26, 9, 13, 26, 26, 8, 13, 13, 7, 13, 13, 26, 8, 13, 14, 26, 8, 14, 14, 9, 14, 14, 26, 8, 14, 14, 4, 14, 14, 14, 15, 5, 15, 9, 9, 15, 9, 5, 5, 15, 9, 5, 15, 9, 15, 9, 9, 15, 9, 5, 15, 9, 5, 15, 9, 9, 15, 9, 9, 15, 10, 9, 25, 7, 7, 15, 8, 9, 25, 7, 15, 8, 15, 8, 9, 16, 8, 9, 16, 8, 6, 24, 7, 16, 8, 16, 8, 6, 16, 8, 9, 16, 9, 6, 16, 9, 6, 16, 7, 7, 16, 7, 7, 9, 16, 7, 7, 16, 7, 16, 7, 16, 7, 9, 9, 16, 7, 9, 16, 8, 16, 8, 9, 6, 16, 8, 6, 7, 16, 8, 7, 7, 7, 5, 16, 8, 7, 7, 17, 8, 7, 7, 5, 17, 8, 9, 7, 7, 7, 17, 8, 7, 7, 9, 8, 17, 8, 7, 5, 5, 17, 8, 7, 17, 8, 7, 17, 8, 7, 9, 9, 17, 8, 9, 17, 9, 7, 9, 5, 17, 9, 7, 9, 7, 17, 9, 7, 9, 9, 7, 17, 9, 7, 8, 8, 17, 9, 7, 8, 17, 9, 7, 8, 17, 9, 7, 8, 8, 8, 17, 9, 8, 17, 9, 7, 8, 7, 21, 21, 7, 17, 9, 8, 8, 5, 8, 8, 17, 9, 8, 5, 17, 9, 8, 8, 8, 17, 9, 8, 8, 17, 9, 8, 17, 9, 8, 5, 17, 9, 8, 8, 8, 8, 17, 9, 8, 5, 17, 9, 8, 8, 17, 9, 8, 17, 9, 8, 8, 17, 9, 9, 20, 17, 8, 8, 9, 8, 8, 8, 8, 8, 17, 9, 9, 20, 17, 8, 8, 20, 17, 9, 20, 17, 9, 20, 17, 9, 8, 20, 17, 10, 20, 17, 10, 8, 20, 17, 17, 10, 17, 10, 8, 20, 7, 17, 10, 8, 20, 7, 18, 10, 18, 10, 8, 20, 7, 18, 10, 20, 7, 18, 10, 18, 10, 8, 18, 10, 8, 20, 7, 18, 10, 18, 9, 8, 8, 20, 8, 18, 9, 8, 18, 9, 8, 18, 9, 8, 8, 20, 8, 18, 9, 9, 18, 9, 4, 18, 9, 4, 10, 8, 19, 8, 18, 9, 18, 9, 4, 19, 8, 18, 9, 18, 9, 5, 19, 8, 18, 9, 18, 9, 8, 19, 8, 19, 8, 8, 19, 8, 19, 8, 18, 9, 8, 19, 8, 19, 7, 18, 9, 8, 19, 8, 20, 8, 5, 20, 8, 20, 8, 18, 9, 8, 20, 8, 20, 8, 8, 20, 8, 20, 8, 20, 8, 20, 10, 10, 20, 10, 4, 5, 20, 10, 5, 10, 10, 20, 10, 5, 10, 20, 10, 5, 20, 10, 5, 5, 20, 10, 5, 10, 20, 10, 5, 10, 20, 10, 5, 5, 20, 10, 5, 5, 20, 11, 20, 3, 5, 17, 11, 7, 7, 20, 11, 8, 5, 17, 11, 7, 20, 11, 8, 20, 11, 8, 5, 20, 9, 8, 6, 20, 9, 8, 9, 17, 8, 7, 20, 9, 8, 20, 9, 8, 9, 20, 9, 8, 6, 20, 9, 6, 9, 20, 9, 6, 9, 20, 9, 7, 7, 20, 10, 7, 8, 6, 20, 10, 8, 8, 20, 10, 8, 20, 10, 8, 20, 10, 8, 6, 6, 20, 10, 8, 6, 20, 10, 8, 20, 10, 8, 6, 10, 20, 10, 8, 10, 8, 20, 10, 8, 8, 8, 8, 10, 20, 10, 8, 8, 8, 20, 8, 8, 8, 8, 10, 20, 8, 8, 6, 8, 9, 9, 20, 9, 8, 9, 9, 6, 8, 20, 9, 9, 9, 10, 10, 20, 9, 9, 9, 20, 9, 9, 9, 20, 9, 9, 9, 6, 6, 20, 9, 9, 6, 20, 9, 6, 9, 6, 10, 20, 9, 6, 10, 6, 10, 20, 10, 6, 10, 6, 6, 10, 20, 10, 6, 10, 9, 9, 20, 10, 6, 10, 9, 20, 10, 7, 10, 9, 20, 10, 7, 10, 9, 9, 9, 20, 10, 7, 9, 20, 10, 7, 10, 9, 10, 15, 20, 15, 20, 10, 20, 10, 7, 9, 9, 10, 9, 9, 20, 10, 7, 9, 10, 20, 10, 7, 9, 9, 9, 20, 11, 7, 9, 10, 20, 11, 7, 10, 20, 11, 7, 10, 10, 20, 11, 7, 10, 10, 10, 10, 20, 11, 7, 10, 10, 20, 11, 7, 10, 10, 20, 11, 7, 10, 20, 11, 7, 10, 10, 20, 11, 7, 7, 14, 2, 20, 11, 10, 10, 7, 10, 10, 10, 10, 10, 20, 11, 7, 7, 14, 2, 20, 11, 10, 11, 14, 2, 20, 11, 7, 14, 2, 20, 11, 7, 14, 2, 20, 11, 7, 10, 14, 2, 20, 11, 20, 3, 14, 3, 20, 11, 20, 3, 10, 14, 3, 20, 11, 20, 11, 20, 3, 20, 11, 20, 3, 10, 14, 3, 11, 20, 11, 20, 3, 10, 14, 3, 11, 20, 10, 20, 3, 20, 10, 20, 3, 11, 14, 3, 11, 20, 11, 20, 3, 14, 3, 11, 20, 11, 20, 3, 20, 11, 20, 3, 11, 20, 11, 20, 3, 11, 14, 3, 11, 20, 11, 20, 4, 20, 11, 7, 11, 11, 14, 4, 11, 20, 11, 7, 11, 20, 11, 7, 11, 20, 11, 7, 11, 11, 14, 4, 11, 20, 11, 7, 7, 20, 11, 7, 6, 20, 11, 7, 6, 20, 4, 11, 20, 7, 12, 20, 12, 7, 21, 12, 7, 6, 21, 8, 12, 21, 12, 8, 21, 12, 8, 8, 21, 8, 12, 21, 12, 8, 21, 12, 8, 12, 21, 8, 12, 21, 8, 12, 12, 21, 8, 12, 21, 8, 12, 21, 12, 8, 12, 21, 8, 12, 21, 8, 10, 21, 12, 8, 12, 21, 8, 12, 13, 4, 12, 8, 13, 4, 12, 13, 4, 12, 21, 12, 8, 12, 13, 4, 13, 13, 4, 13, 13, 13, 4, 13, 13, 4, 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8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 15, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 26, 15, 8, 8, 26, 15, 8, 26, 15, 8, 6, 26, 15, 8, 26, 15, 8, 6, 6, 26, 15, 8, 8, 8, 26, 15, 8, 8, 6, 26, 15, 8, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 8, 8, 26, 15, 8, 8, 6, 26, 15, 8, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 8, 8, 26, 15, 8, 8, 6, 26, 15, 8, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 8, 8, 26, 15, 8, 8, 6, 26, 15, 8, 8, 6, 6, 26, 15, 8, 8, 8, 26, 15, 8, 8, 6, 26, 15, 8, 8, 6, 6, 26, 15, 8, 8, 8, 26, 15, 8, 8, 6, 26, 15, 8, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 8, 8, 26, 15, 8, 8, 6, 26, 15, 8, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 8, 6, 6, 26, 15, 8, 8, 8, 26, 15, 8, 8, 6, 26, 15, 8, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 8, 8, 26, 15, 8, 8, 6, 26, 15, 8, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 8, 8, 26, 15, 8, 8, 6, 26, 15, 8, 8, 6, 6, 26, 15, 8, 8, 8, 26, 15, 8, 8, 6, 26, 15, 8, 8, 6, 6, 26, 15, 8, 8, 8, 26, 15, 8, 8, 6, 26, 15, 8, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 8, 6, 6, 26, 15, 8, 8, 6, 6, 26, 15, 8, 8, 8, 26, 15, 8, 8, 6, 26, 15, 8, 8, 6, 6, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 8, 8, 26, 15, 8, 8, 6, 26, 15, 8, 8, 6, 9, 26, 15, 9, 26, 15, 26, 15, 8, 8, 8, 26, 15, 8, 8, 6, 26, 15, 8, 8, 6,
-------------- next part --------------
from __future__ import division
from math import floor
used = {0:0,
1:0,
2:0,
3:0,
4:0,
5:0,
6:0,
7:0,
8:0,
9:0}
st = "100"
fin = ""
ls = []
pos = 0
while pos < 500000:
if pos == 0:
a = int(st[pos])
else:
a = int(fin[pos])
used[a] += 1
b = pos + 1
c = int(floor(100*used[a]/b))
fin += str(c)
ls.append(c)
pos += 1
with open("newseq.txt","w") as f:
for x in ls:
f.write(str(x)+", ")
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