When d(m) = d(m+n) = n

[Maximilian Hasler]

>  a(n) = smallest positive integer such that
>  d(a(n)) = d(a(n)+2n) = 2n,
>  where d(m) is the number of positive divisors of
>  m.
>
>  The sequence begins: 3,6,12,70,...

lq080419(n,a)={n*=2;until(numdiv(a)==n && numdiv(a+n)==n,a++);a}

for(i=1,99,print1(lq080419(i)", "))
3, 6, 12, 70, 600281, 60, 1458, 264, 450, 266875,
  *** until: user interrupt after 1mn, 18,250 ms.
(next term seems quite large, maybe the function should be made a bit
more intelligent....)
Maximilian




> From seqfan-owner at ext.jussieu.fr  Sat Apr 19 00:34:29 2008
> From: "Eric" <moongerms at wanadoo.fr>
> To: <seqfan at ext.jussieu.fr>, <njas at research.att.com>
> Date: Sat, 19 Apr 2008 00:34:03 +0200
> 
> for
> 7,8,11,12,15,17,19
> we have A112929  a(n) = order of n-th term of A112925 among squarefree
> integers.
> and
> A111105  Number of squarefree integers not exceeding the n-th prime.
> (without crossref write in database between the two sequences)

We have for A112929 the formula

%F A112929 A005117(a(n)) = A112925(n)

and for A112925 the reciprocal formula

%F A112925 a(n) = A005117(A112929(n))

And indeed A111105 is a duplicate of A112929 (where I call the duplicate
the sequence with the later OEIS time stamp).

--
Richard




> From seqfan-owner at ext.jussieu.fr  Sat Apr 19 00:34:29 2008
> From: "Eric" <moongerms at wanadoo.fr>
> To: <seqfan at ext.jussieu.fr>, <njas at research.att.com>
> Subject: A026467|A113626
> Date: Sat, 19 Apr 2008 00:34:03 +0200
> 
> if i take
> A080686
> 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,22,23,24,25,26,27,2
> 7,28,28,29,30,31,32,33,33,34,35,36,36,37,37,38,39,39,39,40,41,42,43,44,44,45
> ,46,47,48,48,48,49,49,49,50,51,52,53,53,54,54,55,55,56
> minus
> A047332
> 0,2,3,5,6,7,9,10,12,13,14,16,17,19,20,21,23,24,26,27,28,30,31,33,34,35,37,38
> ,40,41,42,44,45,47,48,49,51,52,54,55,56,58,59,61,62,63,65,66,68,69,70,72,73,
> 75,76,77,79
> 
> i have
> -1,0,0,1,1,1,2,2,3,3,3,4,4,5,5,5,6,6,7,7,7,8,9,10,10,10,11,11,13,13,14,15,15
> ,16,16,16,18,18,19,19,20,21,22,23,23,24,26,26,27,27,27,28,29,30,30,30,31
> which is funny linked to A085269  Integer part of the conversion from
> centimeters to inches,

This pattern of A047332(n)-A080686(n) = A085269(n-1) fails for n=23 and later;
in detail we have the following table, which shows that these are just
coincidences so far. 

n  A047332(n)-A080686(n)-A085269(n-1)
---------------------------------------
1 -1
2 0
3 0
4 0
5 0
6 0
7 0
8 0
9 0
10 0
11 0
12 0
13 0
14 0
15 0
16 0
17 0
18 0
19 0
20 0
21 0
22 0
23 1
24 1
25 1
26 1
27 1
28 1
29 2
30 2
31 3
32 3
33 3
34 4
35 3
36 3
37 4
38 4
39 5
40 4
41 5
42 5
43 6
44 7
45 6
46 7
47 8
48 8
49 9
50 8
51 8
52 8
53 9
54 10
55 9
56 9
57 9
58 10
59 12
60 11
61 12
62 13
63 13
64 14
65 13
66 13
67 15
68 14
69 16
70 15
71 16
72 17
73 17
74 19
75 18
76 18
77 19
78 18
79 20
80 19
81 19
82 21
83 21
84 22
85 21
86 22
87 24
88 23
89 25
90 24
91 24
92 26
93 26
94 28
95 27
96 27
97 29
98 28
99 29
100 29
101 29
102 30
103 30
104 31
105 31
106 31
107 33
108 32
109 34
110 34
111 34
112 35
113 35
114 36
115 37
116 37
117 38
118 38
119 39
120 39
121 38
122 40
123 40
124 42
125 42
126 41
127 43
128 42
129 44
130 44
131 44
132 45
133 45
134 46
135 46
136 45
137 47
138 48
139 49
140 49
141 49
142 51
143 51
144 51
145 52
146 52
147 53
148 54
149 55
150 55
151 55
152 56
153 56
154 56
155 57
156 56
157 58
158 59
159 60
160 60
161 61
162 61
163 62
164 63
165 63
166 64
167 65
168 65
169 65
170 65
171 65
172 66
173 67
174 68
175 68
176 68
177 69
178 70
179 71
180 71
181 72
182 72
183 73
184 74
185 75
186 76
187 76
188 77
189 77
190 77
191 78
192 78
193 79
194 81
195 80
196 80
197 81
198 81
199 83
200 82

Maple code to produce the table:

isksmooth := proc(n,k)
end:
A080686 := proc(n)
end:
isA047332 := proc(n)
end:
A047332 := proc(n) option remember ;
end:
A085269 := proc(n)
end:

for n from 1 to 200 do
od: