[seqfan] Re: Fractions, digits, decimal expansion

Lars Blomberg larsl.blomberg at comhem.se
Tue Dec 15 16:39:44 CET 2015

```Hello,

Maybe I am misunderstanding something, but I find a(63)=9.

1
6	0.1(6)
13	0.(461538)
10	1.3(0)
14	0.(714285)
17	0.(8235294117647058)
7	2.(428571)
8	0.875(0)
19	0.(421052631578947368)
23	0.(8260869565217391304347)
21	1.0(952380)
29	0.(7241379310344827586206896551)
34	0.8(5294117647058823)
31	1.0(967741935483870)
3	10.(3)
38	0.0(789473684210526315)
28	1.3(571428)
46	0.(6086956521739130434782)
47	0.(9787234042553191489361702127659574468085106382)
35	1.3(428571)
39	0.(897435)
49	0.(795918367346938775510204081632653061224489)
43	1.(139534883720930232558)
51	0.(8431372549019607)
42	1.2(142857)
41	1.0(24390)
48	0.8541(6)
53	0.(9056603773584)
26	2.0(384615)
12	2.1(6)
57	0.(210526315789473684)
58	0.9(8275862068965517241379310344)
59	0.(9830508474576271186440677966101694915254237288135593220338)
2	29.5(0)
61	0.0(327868852459016393442622950819672131147540983606557377049180)
24	2.541(6)
68	0.(3529411764705882)
11	6.(18)
52	0.21(153846)
63	0.(825396)
22	2.8(63)
69	0.(3188405797101449275362)
62	1.1(129032258064516)
71	0.(87323943661971830985915492957746478)
56	1.267(857142)
65	0.8(615384)
76	0.85(526315789473684210)
81	0.(938271604)
44	1.84(09)
67	0.(656716417910447761194029850746268)
64	1.046875(0)
83	0.(77108433734939759036144578313253012048192)
85	0.9(7647058823529411)
78	1.0(897435)
77	1.0(129870)
79	0.(9746835443037)
72	1.097(2)
84	0.(857142)
86	0.(976744186046511627906)
87	0.(9885057471264367816091954022)
80	1.0875(0)
89	0.(89887640449438202247191011235955056179775280)
9	9.(8)

As to the question of the sequence being a permutation,
the following table shows:
C=Number of terms calculated
F=First term that is missing
C	F	F/C
1000	5	0.005
2000	50	0.025
5000	1480	0.296
10000	1650	0.165
20000	2475	0.124
50000	24750	0.495
100000	83250	0.833
200000	198000	0.990
500000	495000	0.990
which seems to support the conjecture.

/Lars

-----Ursprungligt meddelande-----
Från: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] För Hans Havermann
Skickat: den 15 december 2015 14:19
Till: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Ämne: [seqfan] Re: Fractions, digits, decimal expansion

Bob Selcoe:

> I may have spoken too soon:  80190/9 = 8910.  It may be impossible for
> S to equal 80190 or another "precursor" of 9 but I'm not so sure.
>
> Perhaps someone who knows how to program could expand the sequence and
> see if 9 appears?

A no-leading-or-trailing-zeros version of the sequence has 9 appearing after 89, earlier than the appearances of either 4 or 5. Allowing the leading and trailing zeros makes the sequence *less* constrained.

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