request for help in understanding a new sequence (163 line message)

Wed Sep 27 03:36:55 CEST 2006

Dear Seqfans,  Once again I ask for help in understanding
a recent submission.

The sequence in question is:

%I A123031
%S A123031 2,3,3,5,4,5,7,6,6,7,9,8,7,8,11,11,10,9,9,12,13,13,12,11,10,13,14,17,15,
%T A123031 14,13,12,12,15,18,19,17,16,15,14,13,14,19,20,23,19,18,17,16,15,15,16,
%U A123031 21,24,29,21,20,19,18,17,16,17,20,25,30,31,23,22,21,20,19,18,18,21,22
%N A123031 This sequence contains all sequences described by the following where x is a whole number: When divided by all lesser or equal positive integers (greater than 1) the number contains only 1 remainder of x.
%O A123031 1,1
%K A123031 nonn
%A A123031 Jared B. Ricks (jaredricks(AT)yahoo.com), Sep 24 2006

I asked the author for further details. This is his reply.
But I am still confused. 
Could someone kindly send me a simpler description?
Thanks!
Neil Sloane

(start of his reply)

This is what the sequence looks like when formatted as a triangle:
x)
0) 2, 3, 5, 7, 11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,
1) 3, 4, 6, 8, 12,14,18,20,24,30,32,38,42,44,48,54,60,62,68,72,74,80,84,90,98,
2) 5, 6, 7, 9, 13,15,19,21,25,31,33,39,43,45,49,55,61,63,69,73,75,81,85,91,99,
3) 7, 8, 9, 10,12,14,16,20,22,26,32,34,40,44,46,50,56,62,64,70,74,76,82,86,92,
4) 9, 10,11,12,13,15,17,21,23,27,33,35,41,45,47,51,57,63,65,71,75,77,83,87,93,
5) 11,12,13,14,15,16,18,20,22,24,28,30,34,36,42,46,48,52,58,64,66,72,76,78,84,
6) 13,14,15,16,17,18,19,21,23,25,29,31,35,37,43,47,49,53,59,65,67,73,77,79,85,
7) 15,16,17,18,19,20,21,22,24,26,28,30,32,36,38,42,44,48,50,54,56,60,66,68,74,
8) 17,18,19,20,21,22,23,24,25,27,29,31,33,37,39,43,45,49,51,55,57,61,67,69,75,
9) 19,20,21,22,23,24,25,26,27,28,30,32,34,36,38,40,44,46,50,52,56,58,62,68,70,
10) 21,22,23,24,25,26,27,28,29,30,31,33,35,37,39,41,45,47,51,53,57,59,63,69,71,
11) 23,24,25,26,27,28,29,30,31,32,33,34,36,38,40,42,44,46,48,52,54,58,60,64,66,
12) 25,26,27,28,29,30,31,32,33,34,35,36,37,39,41,43,45,47,49,53,55,59,61,65,67,
13) 27,28,29,30,31,32,33,34,35,36,37,38,39,40,42,44,46,48,50,52,54,56,60,62,66,
14) 29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,45,47,49,51,53,55,57,61,63,67,
15) 31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,48,50,52,54,56,58,60,62,64,
16) 33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,51,53,55,57,59,61,63,65,
17) 35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,54,56,58,60,62,64,66,
18) 37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,57,59,61,63,65,67,
19) 39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,60,62,64,66,68,

I describe each row in the following manner:
When divided by all lesser or equal positive integers 
(greater than 1) the number contains only 1 remainder of x.

So, for example, if I wanted to get row 2, the description would be:

When divided by all lesser or equal positive integers 
(greater than 1) the number contains only 1 remainder of 2.

I would obtain row 2 by creating and examining the following chart:

	2  3  4  5  6  7  8  9 10  11  12  13 ...

-6)	0, 0, 2, 4, 0, 1, 2, 3, 4,  5,  6,  7, ...
-5)	1, 1, 3, 0, 1, 2, 3, 4, 5,  6,  7,  8, ... (This Row Contains only one 2)
-4)	0, 2, 0, 1, 2, 3, 4, 5, 6,  7,  8,  9, ... 
-3)	1, 0, 1, 2, 3, 4, 5, 6, 7,  8,  9, 10, ... (This Row Contains only one 2)
-2)	0, 1, 2, 3, 4, 5, 6, 7, 8,  9, 10, 11, ... (This Row Contains only one 2)
-1)	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, ... (This Row Contains only one 2)	
 0)	0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0, ... 
 1)     1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1,  1, ...
 2)	0, 2, 2, 2, 2, 2, 2, 2, 2,  2,  2,  2, ... This is the middle row (This is because it is row 2)
 3)	1, 0, 3, 3, 3, 3, 3, 3, 3,  3,  3,  3, ... 
 4)	0, 1, 0, 4, 4, 4, 4, 4, 4,  4,  4,  4, ...
 5)     1, 2, 1, 0, 5, 5, 5, 5, 5,  5,  5,  5, ...  (This Row Contains only one 2)
 6)	0, 0, 2, 1, 0, 6, 6, 6, 6,  6,  6,  6, ... (This Row Contains only one 2)
 7)	1, 1, 3, 2, 1, 0, 7, 7, 7,  7,  7,  7, ... (This Row Contains only one 2)
 8)	0, 2, 0, 3, 2, 1, 0, 8, 8,  8,  8,  8, ...
 9)	1, 0, 1, 4, 3, 2, 1, 0, 9,  9,  9,  9, ... (This Row Contains only one 2)
10)	0, 1, 2, 0, 4, 3, 2, 1, 0, 10, 10, 10, ...

Now, collect the row number of each row that contains
only one 2.
 From the partial chart above, we get:
-5,-3,-2,-1,5,6,7,9
Since there exists a perfect symmetry around x (in our
case x is 2),
there is no need to include the negative numbers. 
This leaves us with 
the following:
5,6,7,9,...

If we repeat this process for all numbers 'x' we get
the following chart:

x)
0)	2, 3, 5, 7,
11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,
1)	3, 4, 6, 8,
12,14,18,20,24,30,32,38,42,44,48,54,60,62,68,72,74,80,84,90,98,
2)	5, 6, 7, 9,
13,15,19,21,25,31,33,39,43,45,49,55,61,63,69,73,75,81,85,91,99,
3)	7, 8, 9,
10,12,14,16,20,22,26,32,34,40,44,46,50,56,62,64,70,74,76,82,86,92,

4)	9,
10,11,12,13,15,17,21,23,27,33,35,41,45,47,51,57,63,65,71,75,77,83,87,93,
5)
11,12,13,14,15,16,18,20,22,24,28,30,34,36,42,46,48,52,58,64,66,72,76,78,84,

6)
13,14,15,16,17,18,19,21,23,25,29,31,35,37,43,47,49,53,59,65,67,73,77,79,85,

7)
15,16,17,18,19,20,21,22,24,26,28,30,32,36,38,42,44,48,50,54,56,60,66,68,74,

8)
17,18,19,20,21,22,23,24,25,27,29,31,33,37,39,43,45,49,51,55,57,61,67,69,75,

9)
19,20,21,22,23,24,25,26,27,28,30,32,34,36,38,40,44,46,50,52,56,58,62,68,70,

10)
21,22,23,24,25,26,27,28,29,30,31,33,35,37,39,41,45,47,51,53,57,59,63,69,71,

11)
23,24,25,26,27,28,29,30,31,32,33,34,36,38,40,42,44,46,48,52,54,58,60,64,66,

12)
25,26,27,28,29,30,31,32,33,34,35,36,37,39,41,43,45,47,49,53,55,59,61,65,67,

13)
27,28,29,30,31,32,33,34,35,36,37,38,39,40,42,44,46,48,50,52,54,56,60,62,66,

14)
29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,45,47,49,51,53,55,57,61,63,67,

15)
31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,48,50,52,54,56,58,60,62,64,

16)
33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,51,53,55,57,59,61,63,65,

17)
35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,54,56,58,60,62,64,66,

18)
37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,57,59,61,63,65,67,

19)
39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,60,62,64,66,68,

In order to combine it all into one sequence, I read
the numbers diagonally starting 
with the '2' on row '0'. So we get:

2, 3, 3, 5, 4, 5, 7, 6, 6, 7, 9, 8, 7, 8, 11, ...

Thanks for your interest.  I appologize for not being
so clear in the beginning.

Sincerely,

Jared Ricks
jaredricks at yahoo.com

End)