[seqfan] Re: A mysterious sequence from Russia
David Applegate
david at research.att.com
Fri Feb 28 23:49:50 CET 2014
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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