[seqfan] Re: A mysterious sequence from Russia

[Neil Sloane]

Alan, Dave,

it is beginning to smell quite sweet ...

i'll create two new sequences (numerators and denoms) for it tomorrow

Thanks for the help!

Neil


On Fri, Feb 28, 2014 at 5:49 PM, David Applegate <david at research.att.com>wrote:

> I believe that, from the example and some experimenting, I can explain
> the underlying sequence.  It gives, for primes p >= 7, the average
> digit value in the periodic portion of the decimal expansion of 1/p.
>
> Hence,
>
>       average
>        digit    sequence
>   N    value     terms
>   7     9/2       4,5      0.(142857)*
>  11     9/2       4,5      0.(09)*
>  13     9/2       4,5      0.(076923)*
>  17     9/2       4,5      0.(0588235294117647)*
>  19     9/2       4,5      0.(052631578947368421)*
>  23     9/2       4,5      0.(0434782608695652173913)*
>  29     9/2       4,5      0.(0344827586206896551724137931)*
>  31    18/5       3,6      0.(032258064516129)*
>  37     3         3,0      0.(027)*
>  41    18/5       3,6      0.(02439)*
>  43    30/7       4,285700  note 30/7=4.(285714)*
>  47     9/2       4,5
>  53    63/13      4,846200  note 63/13=4.(846153)*
>  59     9/2       4,5
>  61     9/2       4,5
>  67    48/11      4,363600  note 48/11=4.(36)*
>  71    18/5       3,6
>  73     9/2       4,5
>  79    54/13      4,153800  note 54/13=4.(153846)*
>  83   171/41      4,170700  note 171/41=4.(17073)*
>  89     9/2       4,5
> ...
>
> The english discussion about odd/even is, I believe, that if the
> period of the periodic decimal expansion of 1/p is even, then the
> value is 9/2.  The period always divides p-1.
>
> Related existing sequences include A002371 (period of decimal
> expansion of 1/(n-th prime)), and A060283 (periodic part of decimal
> expansion of recipricol of n-th prime).
>
> The rational version of this sequence is
> digit_sum(A060283(n))/A002371(n).
>
> A maple function to compute the n-th term of the rational version of
> this sequence (starting with n=4) is:
>
> A := proc(n) local i,p;
>    p := ithprime(n);
>    add(i,i=convert((10^(p-1)-1)/p,base,10))/(p-1);
> end proc;
>
> A hideous maple function to compute the n-th term of the sequence in
> the dumpster is:
>
> B := proc(n) local i,j;
>    if n mod 2 = 1 then floor(A((n+1)/2+3));
>    else
>       i := A(n/2+3):
>       j := i - floor(i):
>       if j = floor(j*10)/10 then j*10;
>       else round(j*10000)*100;
>       end if;
>    end if;
> end proc;
>
> However, even though I dived into the dumpster to figure out what this
> is, I am too appalled by this trash to be willing to enter it into the
> OEIS.  If someone else values some version of it enough, please go ahead.
>
> -Dave
>
> > From seqfan-bounces at list.seqfan.eu Fri Feb 28 13:57:12 2014
> > Date: Fri, 28 Feb 2014 13:56:19 -0500
> > From: Neil Sloane <njasloane at gmail.com>
> > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> > Subject: [seqfan] A mysterious sequence from Russia
>
> > I have a friend who collects abandoned
> > computers from the town dump. I sometimes look for sequences in the OEIS
> > trash heap of abandoned sequences.
>
> > Here is one such:
>
> > Question: what is this sequence?
>
> > 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 3, 6, 3, 0, 3, 6, 4, 285700, 4,
> > 5, 4, 846200, 4, 5, 4, 5, 4, 363600, 3, 6, 4, 5, 4, 153800, 4, 170700, 4,
> > 5, 4, 5, 4, 5, 4, 5, 4, 245300, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5
>
> > Source: an abandoned version of A230604 (see the "history")
>
> > Hints (from the "history"):
>
> > 1/N = 0,(a_1a_2a_3a...a_n)
>
> > k= S/n
>
> > There are two kinds of N. The first one is: the length of the period is
> an
> > even and k is 4.5. The second- where the length is odd.
>
> > The improve:Пусть 1/p = 0,a_1a_2a_3...И пусть у нас есть период
> > (a_1a_2a_3...a_k)Поймём, что число a_1a_2...a_k (уже натуральное, без
> > ведущей десятичной запятой) равно (10^n - 1) / p для первого    ... (long
> > text follows)
>
> > Example: If N is 7,1/7=0.(142857)
>
> > S=1+4+2+8+5+7=27
>
> > n=6
>
> > k=4.5
>
> > This suggests that this is really a sequence of fractions: 9/2, 9/2, ....
> > but what is the real definition?
>
> > This may or may not be an interesting sequence. Maybe someone who reads
> > Russian could take a look . This is such a classical part of elementary
> > number theory that it is unlikely to be new. But one never knows.
>
> > Neil
>
> > _______________________________________________
>
> > Seqfan Mailing list - http://list.seqfan.eu/
>
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>
>


-- 
Dear Friends, I have now retired from AT&T. New coordinates:

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