Corrections wanted to extend sequence A035335

[Jonathan Post]

I've taken a few minutes to extend A035335, but am not sure of my results.

A035335  Smallest number such that n appears in decimal digits of 1/a(n).

n   a(n)
77 36
78 19 because 1/19 = 0.0526315789... and the "78" is 8 and 9 digits
right of point
79 53
80 31
81 43
82 29
83 12
84 19
85 14
86 23
87 47
88 17
89 19
90 11
91 46
92 13
93 58
94 17
95 23
96 29
97 47
98 59
99 101

One can also demand that the a(n) be prime, which gives a different
sequence.  For instance 1/83 is the smallest reciprocal of a prime
whose decimal digits include "77" as a substring, while the composite
reciprocal 1/36 is the smallest with that same substring.

For 83, these are 53 prime, 12 composite
For 85: 47 prime, 14 composite
For 91, 47 prime, 46 composite, and so forth.

My play with this was "by hand" -- might someone want to code this,
and make a b-list?

Also, in the usual sense, base 10 is arbitrary.  What do we know about
the base k analogue of A035335 for other k?




jvp> From seqfan-owner at ext.jussieu.fr  Sun Jul 13 18:18:15 2008
jvp> Date: Sun, 13 Jul 2008 09:17:06 -0700
jvp> From: "Jonathan Post" <jvospost3 at gmail.com>
jvp> To: SeqFan <seqfan at ext.jussieu.fr>
jvp> Subject: Corrections wanted to extend sequence A035335
jvp> 
jvp> I've taken a few minutes to extend A035335, but am not sure of my results.
jvp> 
jvp> A035335  Smallest number such that n appears in decimal digits of 1/a(n).
jvp> 
jvp> n   a(n)
jvp> 77 36
jvp> 78 19 because 1/19 = 0.0526315789... and the "78" is 8 and 9 digits
jvp> right of point
jvp> 79 53

I get 79 29, because (location marked by parentheses)
1/29 = 0.034482758620689655172413(79)31...

jvp> 80 31
jvp> 81 43
jvp> 82 29

I get an earlier 82 17, from 1/17 = 0.058(82)3529411764...

jvp> 83 12
jvp> 84 19
jvp> 85 14

85 7, from 1/7 = 0.142(85)7142...

jvp> 86 23
jvp> 87 47
jvp> 88 17
jvp> 89 19
jvp> 90 11
jvp> 91 46

91 23 from 1/23 = 0.0434782608695652173(91)304...

jvp> 92 13
jvp> 93 58

93 29 from 1/29 = 0.0344827586206896551724137(93)10...

jvp> 94 17
jvp> 95 23
jvp> 96 29
jvp> 97 47
jvp> 98 59
jvp> 99 101

My first 200 terms are
1 6
2 4
3 3
4 7
5 2
6 6
7 7
8 7
9 11
10 10
11 9
12 8
13 23
14 7
15 19
16 6
17 17
18 49
19 21
20 5
21 19
22 31
23 13
24 29
25 4
26 19
27 29
28 7
29 17
30 13
31 19
32 31
33 3
34 23
35 17
36 19
37 27
38 26
39 23
40 25
41 17
42 7
43 23
44 29
45 22
46 26
47 17
48 29
49 51
50 2
51 29
52 17
53 26
54 22
55 18
56 23
57 7
58 17
59 47
60 23
61 21
62 16
63 19
64 17
65 23
66 6
67 49
68 19
69 13
70 17
71 7
72 29
73 19
74 47
75 29
76 13
77 36
78 19
79 29
80 31
81 43
82 17
83 12
84 19
85 7
86 23
87 47
88 17
89 19
90 11
91 23
92 13
93 29
94 17
95 23
96 29
97 47
98 59
99 101
100 10
101 59
102 39
103 29
104 96
105 19
106 47
107 93
108 83
109 91
110 109
111 9
112 71
113 88
114 61
115 113
116 86
117 17
118 59
119 84
120 83
121 82
122 49
123 81
124 129
125 8
126 71
127 47
128 78
129 31
130 23
131 61
132 83
133 75
134 67
135 59
136 73
137 29
138 72
139 43
140 71
141 113
142 7
143 167
144 69
145 131
146 109
147 61
148 47
149 67
150 106
151 66
152 59
153 26
154 97
155 109
156 64
157 19
158 63
159 113
160 131
161 31
162 86
163 49
164 97
165 109
166 6
167 131
168 113
169 59
170 47
171 157
172 29
173 23
174 109
175 57
176 17
177 107
178 56
179 89
180 61
181 55
182 181
183 49
184 179
185 54
186 59
187 133
188 53
189 179
190 21
191 47
192 52
193 129
194 113
195 82
196 51
197 157
198 131
199 201

Richard Mathar




I think that a=37 is missing in A005360 (and therefore
to be removed from A125121) because with k=7085 we
have the binary representations
37=[1, 0, 1, 0, 0, 1]
37*k= [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]

The same problem occurs with 74, 83, and 101, all to be added to A005360
and to be removed from A125121.

The main question is: is there an algorithm to demonstrate
absence of numbers in A005360? (The above were found by a
brute-force scan with some arbitrary upper k=30000.)

Richard Mathar
K. Stolarsky, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa38/aa3825.pdf">Intgers whose multiples have anomlaous digital frequencies</a>, Acta Arith. 38 (1981) 117-128

two binary representations of a and a*k are (not necesarily complete, as argued!):

11 3 [1, 1, 0, 1] [1, 0, 0, 0, 0, 1]
13 5 [1, 0, 1, 1] [1, 0, 0, 0, 0, 0, 1]
19 27 [1, 1, 0, 0, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0, 1]
22 3 [0, 1, 1, 0, 1] [0, 1, 0, 0, 0, 0, 1]
23 3 [1, 1, 1, 0, 1] [1, 0, 1, 0, 0, 0, 1]
25 41 [1, 0, 0, 1, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
26 5 [0, 1, 0, 1, 1] [0, 1, 0, 0, 0, 0, 0, 1]
27 3 [1, 1, 0, 1, 1] [1, 0, 0, 0, 1, 0, 1]
29 5 [1, 0, 1, 1, 1] [1, 0, 0, 0, 1, 0, 0, 1]
37 7085 [1, 0, 1, 0, 0, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
38 27 [0, 1, 1, 0, 0, 1] [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1]
39 7 [1, 1, 1, 0, 0, 1] [1, 0, 0, 0, 1, 0, 0, 0, 1]
41 25 [1, 0, 0, 1, 0, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
43 3 [1, 1, 0, 1, 0, 1] [1, 0, 0, 0, 0, 0, 0, 1]
44 3 [0, 0, 1, 1, 0, 1] [0, 0, 1, 0, 0, 0, 0, 1]
46 3 [0, 1, 1, 1, 0, 1] [0, 1, 0, 1, 0, 0, 0, 1]
47 3 [1, 1, 1, 1, 0, 1] [1, 0, 1, 1, 0, 0, 0, 1]
50 41 [0, 1, 0, 0, 1, 1] [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
52 5 [0, 0, 1, 0, 1, 1] [0, 0, 1, 0, 0, 0, 0, 0, 1]
53 5 [1, 0, 1, 0, 1, 1] [1, 0, 0, 1, 0, 0, 0, 0, 1]
54 3 [0, 1, 1, 0, 1, 1] [0, 1, 0, 0, 0, 1, 0, 1]
55 3 [1, 1, 1, 0, 1, 1] [1, 0, 1, 0, 0, 1, 0, 1]
57 9 [1, 0, 0, 1, 1, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0, 1]
58 5 [0, 1, 0, 1, 1, 1] [0, 1, 0, 0, 0, 1, 0, 0, 1]
59 3 [1, 1, 0, 1, 1, 1] [1, 0, 0, 0, 1, 1, 0, 1]
61 5 [1, 0, 1, 1, 1, 1] [1, 0, 0, 0, 1, 1, 0, 0, 1]
71 119 [1, 1, 1, 0, 0, 0, 1] [1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]
74 7085 [0, 1, 0, 1, 0, 0, 1] [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
76 27 [0, 0, 1, 1, 0, 0, 1] [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1]
77 5 [1, 0, 1, 1, 0, 0, 1] [1, 0, 0, 0, 0, 0, 0, 1, 1]
78 7 [0, 1, 1, 1, 0, 0, 1] [0, 1, 0, 0, 0, 1, 0, 0, 0, 1]
79 7 [1, 1, 1, 1, 0, 0, 1] [1, 0, 0, 1, 0, 1, 0, 0, 0, 1]
82 25 [0, 1, 0, 0, 1, 0, 1] [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
83 395 [1, 1, 0, 0, 1, 0, 1] [1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
86 3 [0, 1, 1, 0, 1, 0, 1] [0, 1, 0, 0, 0, 0, 0, 0, 1]
87 3 [1, 1, 1, 0, 1, 0, 1] [1, 0, 1, 0, 0, 0, 0, 0, 1]
88 3 [0, 0, 0, 1, 1, 0, 1] [0, 0, 0, 1, 0, 0, 0, 0, 1]
91 3 [1, 1, 0, 1, 1, 0, 1] [1, 0, 0, 0, 1, 0, 0, 0, 1]
92 3 [0, 0, 1, 1, 1, 0, 1] [0, 0, 1, 0, 1, 0, 0, 0, 1]
94 3 [0, 1, 1, 1, 1, 0, 1] [0, 1, 0, 1, 1, 0, 0, 0, 1]
95 3 [1, 1, 1, 1, 1, 0, 1] [1, 0, 1, 1, 1, 0, 0, 0, 1]
99 11 [1, 1, 0, 0, 0, 1, 1] [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1]
100 41 [0, 0, 1, 0, 0, 1, 1] [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
101 365 [1, 0, 1, 0, 0, 1, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1]
103 5 [1, 1, 1, 0, 0, 1, 1] [1, 1, 0, 0, 0, 0, 0, 0, 0, 1]
104 5 [0, 0, 0, 1, 0, 1, 1] [0, 0, 0, 1, 0, 0, 0, 0, 0, 1]
106 5 [0, 1, 0, 1, 0, 1, 1] [0, 1, 0, 0, 1, 0, 0, 0, 0, 1]
107 3 [1, 1, 0, 1, 0, 1, 1] [1, 0, 0, 0, 0, 0, 1, 0, 1]
108 3 [0, 0, 1, 1, 0, 1, 1] [0, 0, 1, 0, 0, 0, 1, 0, 1]
109 5 [1, 0, 1, 1, 0, 1, 1] [1, 0, 0, 0, 0, 1, 0, 0, 0, 1]
110 3 [0, 1, 1, 1, 0, 1, 1] [0, 1, 0, 1, 0, 0, 1, 0, 1]
111 3 [1, 1, 1, 1, 0, 1, 1] [1, 0, 1, 1, 0, 0, 1, 0, 1]
113 145 [1, 0, 0, 0, 1, 1, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
114 9 [0, 1, 0, 0, 1, 1, 1] [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1]
115 9 [1, 1, 0, 0, 1, 1, 1] [1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1]
116 5 [0, 0, 1, 0, 1, 1, 1] [0, 0, 1, 0, 0, 0, 1, 0, 0, 1]
117 5 [1, 0, 1, 0, 1, 1, 1] [1, 0, 0, 1, 0, 0, 1, 0, 0, 1]
118 3 [0, 1, 1, 0, 1, 1, 1] [0, 1, 0, 0, 0, 1, 1, 0, 1]
119 3 [1, 1, 1, 0, 1, 1, 1] [1, 0, 1, 0, 0, 1, 1, 0, 1]
121 9 [1, 0, 0, 1, 1, 1, 1] [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1]
122 5 [0, 1, 0, 1, 1, 1, 1] [0, 1, 0, 0, 0, 1, 1, 0, 0, 1]
123 3 [1, 1, 0, 1, 1, 1, 1] [1, 0, 0, 0, 1, 1, 1, 0, 1]
125 5 [1, 0, 1, 1, 1, 1, 1] [1, 0, 0, 0, 1, 1, 1, 0, 0, 1]
139 59 [1, 1, 0, 1, 0, 0, 0, 1] [1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
141 581 [1, 0, 1, 1, 0, 0, 0, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1]
142 119 [0, 1, 1, 1, 0, 0, 0, 1] [0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]
143 15 [1, 1, 1, 1, 0, 0, 0, 1] [1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1]
145 113 [1, 0, 0, 0, 1, 0, 0, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
147 7 [1, 1, 0, 0, 1, 0, 0, 1] [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1]
148 7085 [0, 0, 1, 0, 1, 0, 0, 1] [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
149 55 [1, 0, 1, 0, 1, 0, 0, 1] [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]

The list above is constructed with the naive:
A000120 := proc(n)
end:
isA005360 :=  proc(n)
end:
for n from 1 to 150 do
od: