[seqfan] Re: A mysterious sequence from Russia

[David Applegate]

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:ðÕÓÔØ 1/p = 0,a_1a_2a_3...é ÐÕÓÔØ Õ ÎÁÓ ÅÓÔØ ÐÅÒÉÏÄ
> (a_1a_2a_3...a_k)ðÏÊÍ£Í, ÞÔÏ ÞÉÓÌÏ a_1a_2...a_k (ÕÖÅ ÎÁÔÕÒÁÌØÎÏÅ, ÂÅÚ
> ×ÅÄÕÝÅÊ ÄÅÓÑÔÉÞÎÏÊ ÚÁÐÑÔÏÊ) ÒÁ×ÎÏ (10^n - 1) / p ÄÌÑ ÐÅÒ×ÏÇÏ    ... (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

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