Review: A048859

[David Wilson]

----- Original Message -----
From: "Dean Hickerson" <dean at math.ucdavis.edu>
To: <seqfan at ext.jussieu.fr>
Sent: Thursday, May 30, 2002 6:55 PM
Subject: Re: Review: A048859


> David Wilson asked:
>
> > Is anyone able to relate the values of A048859 to the formula?
>
> Here are the values and 'formula':
>
> %S A048859 1,2,4,7,9,14,20,25,31,34,44
> %F A048859 Keep first k numbers, skip the k+1 numbers, for k=2,3,4,...

I think the only reasonable interpretation of this seive is that we start
with an empty
"processed sequence" and an "unprocessed sequence" consisting of the
positive
integers.  To "keep firsty k numbers" would mean "Transfer k numbers from
the
head of the unprocessed sequence to the processed sequence".  To "skip the
k+1
numbers" might mean:

(1) Drop k+1 numbers from the head of the unprocessed sequence.  This leads
to

1 2 6 7 8 13 14 15 16 22 23 24 25 26 33 34 35 36 37 38 46 47 48 49 50
51 52 61 62 63 64 65 66 67 68 78 79 80 81 82 83 84 85 86 97 98 99 100
101 102 103 104 105 106 118 119 120 121 122 123 124 125 126 127 128 141

(2) Remove every (k+1)st element from the unprocessed sequence, starting at
the (k+1)st element.

1 2 3 4 6 7 9 10 13 15 16 19 21 25 27 28 31 34 37 40 45 46 49 51 57 58
64 67 70 75 76 79 87 88 94 97 100 106 109 111 117 121 127 135 136 139
141 151 154 157 160 169 171 181 184 190 195 196 207 211 214 217 225 226

(3) Remove every (k+1)st element from the unprocessed sequence, starting at
the first element.

1 2 4 5 7 10 11 13 16 19 22 23 25 29 34 35 40 41 43 49 53 55 59 64 65
70 73 82 83 85 89 95 100 101 109 115 119 124 125 130 139 142 143 149 160
163 169 172 173 179 185 191 193 203 209 214 223 229 232 233 235 250 251

(4) Remove every multiple of k+1 from the unprocessed sequence.

1 2 4 5 7 10 11 13 14 17 19 22 23 26 29 31 34 37 38 41 43 46 47 53 58
59 61 62 67 71 73 74 79 82 83 86 89 94 97 101 103 106 107 109 113 118
121 122 127 131 134 137 139 142 146 149 151 157 158 163 166 167 169 173

None of these interpretations result in the posted sequence.

> Since so few terms are given, it seems likely that they were computed by
> hand, and maybe the author miscomputed.  Consider the following
interpretation
> of the formula:  Start with the sequence of positive integers.  Delete
> every 3rd term, then delete every 4th term of what's left, then delete
every
> 5th term, ...  We get:
>
> %S A000000
1,2,4,7,10,14,20,25,32,40,46,55,67,74,91,104,112,127,145,154,175,194,
> %T A000000
200,230,241,265,284,307,325,355,382,404,422,464,475,505,547,556,595,
> %U A000000
631,661,680,736,742,790,847,860,904,950,967,1012,1081,1094,1135,1184
>
> %t A000000 del[lst_,k_]:=lst[[Select[Range[Length[lst]],Mod[#,k]!=0&]]];
For[k=3;s=Range[1000],k<=Length[s],k++,s=del[s,k]]; s
>
> Perhaps the author made 2 mistakes:  He wrote 9 instead of 10, and he
skipped
> the step in which every 9th term is deleted.  After deleting every 3rd
term,
> ..., every 8th term, we get:
>
>     1 2 4 7 10 14 20 25 31 32 34 40 44
>
> Then deleting the 10th term gets rid of 32 and deleting the 11th remaining
> term gets rid of 40, leaving:
>
>     1 2 4 7 10 14 20 25 31 34 44
>
> which agrees with the original sequence except for the 10.
>
> Note that if we instead start by deleting every 2nd term, we get A000960.
>
> Dean Hickerson
> dean at math.ucdavis.edu

Barring input from the author, my inclination is to kill this sequence as
indecipherable.
Any other opinions?