**First digit of a(n)**

... is not the a(n)^{th} digit of S

(this message appeared
more or less like this

on the Seqfan Mailing list
on February 21^{st}, 2010)

Hello Seqfans,

... computed by hand, here is the first (I guess)
lexicographically reordering S
of the Natural numbers where every a(n) says:

- "My first
digit is _not_ the a(n)^{th}
digit of S":

S = 2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 20, 22, 23, 24, 11, 13, 14, 15, 16, 12, 17,
18, 19, 21, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, ...

Example:

10 says that the
10^{th} digit of S
is not 1 -- which is true (its 0)

9 says that the 9^{th} digit of S is not 9 -- which is
true (its 1)

20 says that the
20^{th} digit of S
is not 2 -- which is true (its 1)

...

The
building method is simple: we must prolong S with the smallest unused integer which has no
digit matching the corresponding digit of _the 3^{rd} line_ of the
array below. In the 4^{th} line, read vertically means that the digit count must be red vertically -- in grey or **bold**. For example, this portion of the
array means:

__|12|__
<-- herunder is the 12^{th} integer of S

__,22____,__ <-- this is the 12^{th}
integer of S

**1**1
<-- **12**^{th}
and

__2____3__ <-- 13^{th} digit of S - here the 12^{th} digit of S is 2 and the 13^{th}
is also 2)

**N**
= __1|2|3|4|5|6|7|8| 9|10|11|12|13|14|15|16|17|18|19|20|21|22|23|24|25|26|27|28|29|30|31|32|33|34|35|36|37__|...

-----> S = __2,1,4,3,6,5,8,7,10,9,
20,22,23,24,11,13,14,15,16,12,17,18,19,21,25,26,27,28,29,30,31,32,33,34,35,37,38__,...

S digit# = 1 2 3 4 5 6 7 8 9**1 **1 **1**1 **1**1 **1**1 **1**1 **2**2 **2**2
**2**2 **2**2 **2**2 **3**3 **3**3
**3**3 **3**3 **3**3 **4**4 **4**4
**4**4 **4**4 **4**4 **5**5 **5**5
**5**5 **5**5 **5**5 **6**6 **6**6
**6**6** ...**

S digit# = __r ead__

_____

**Ray Chandler** has just checked the S sequence above and computed
100 terms; here they are:

S = 2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 20, 22, 23, 24, 11, 13, 14, 15, 16, 12,
17, 18, 19, 21, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 36,
41, 42, 43, 44, 45, 46, 47, 49, 50, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83,
84, 85, 86, 87, 88, 89, 90, 92, 93, 94, 95, 96, 97, 98, 99, 200, 202, 203, 204,
205,...

__________

**BTW**:

Two different
sequences based on the same idea (but dropping the re-ordering of the
Naturals constraint from above):

T is the first lexicographically and monotonically increasing seq. where the
1^{st} digit of a(**n**) is not the a(**n**)^{th} digit of T:

**n**
= __1|2|3|4|5|6|7|8| 9|10|11|12|13|14|15|16|17|18|19|20|21|22|23|24|25|26|27|28|29|30|31|32|33|34|35|36__|...

T = a(**n**)
= __2,3,4,5,6,7,8,9,10,12,14,16,18,20,23,30,31,32,34,40,41,42,43,45,50,51,52,53,54,60,61,62,63,64,70,71__,...

T digit# = r**e**a**d**.**v**e**r**t**i**c**a**l**l**y** .1** 1**1** 1**1** 1**1**
1**1** 1**2** 2**2** 2**2**** **2**2**** **2**2**** **2**3 **3**3** 3**3** 3**3** 3**3**
3**4** 4**4** 4**4** 4**4** 4**4**
4**5** 5**5** 5**5** 5**5** 5**5**
5**6** 6**6** 6**6**** ...**

T digit# = __1 2 3 4 5 6 7 8 9 0 12 34 56 78 90 12 34 56 78 90 12 34 56 78 90 12 34 56 78 90 12 34 56 78 90 12 34 ...__

__Exam__p__le__:

for n=1, a(n)=2 whose
1^{st} digit 2 is not the #2 digit of T
-- which is a 3

for n=2, a(n)=3 whose
1^{st} digit 3 is not the #3 digit of T
-- which is a 4

for n=3, a(n)=4 whose
1^{st} digit 4 is not the #4 digit of T
-- which is a 5

...

for n=8, a(n)=9
whose 1^{st} digit 9 is not the #9 digit of T -- which is a 1 -- the 1 from 10

for n=9, a(n)=10 whose
1^{st} digit 1 is not the #**10**
digit of T -- which
is a 0 -- the 0 from 10

for n=10, a(n)=12
whose 1^{st} digit 1 is not the #**12**
digit of T -- which
is a 2 -- the 2 from 12

for n=11, a(n)=14
whose 1^{st} digit 1 is not the #**14**
digit of T -- which
is a 4 -- the 4 from 14

...

for n=20, a(n)=40
whose 1^{st} digit 4 is not the #**40**
digit of T -- which
is a 5 -- the 5 from 45

... so the 1^{st} digit of a(n) is never the a(n)^{th} digit of T.

We see here that
11 is forbidden in T;
11 would momentarily replace 12 in T -- but then 11, saying the 11^{th} digit of T is not
a 1 would lie -- as the 11^{th} digit of T would be 1, precisely --
the 1 of 11. Putting 12 instead of 11 moves the cursor towards the 2
of 12 -- which digit is indeed different from the 1 of 12. As T is monotonically
increasing, 11 will never appear because of the presence of 12.

_____

**Ray Chandler** has just corrected my T sequence above and computed
100 terms; here they are:

T = 2, 3, 4, 5, 6, 7,
8, 9, 10, 12, 14, 16, 18, 20, 23, 30, 31, 32, 34, 40, 41, 42, 43, 45, 50, 51,
52, 53, 54, 60, 61, 62, 63, 64, 70, 71, 72, 73, 74, 80, 81, 82, 83, 84, 90, 91,
92, 93, 94, 100, 101, 102, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209,
210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225,
226, 227, 228, 229, 230, 231, 232, 233, 300, 301, 303, 304, 305, 306, 307, 308,
309, 310, 311, 312, 313, 314,...

_____

If we drop the
monotonically increasing requirement of T, we get U:

U is the first lexicographically seq. where the 1^{st} digit of a(**n**) is not the
a(**n**)^{th}
digit of U:

**n**
= __1|2|3|4|5|6|7|8|
9|10|11|12|13|14|15|16|17|18|19|20|21|22|23|24|25|26|27|28|29|30|31|32|33|34|35|36__|...

U = a(**n**) = __2,1,4,3,6,5,8,7,10,
9,11,15,17,19,21,23,25,27,29,31,34,40,33,35,39,41,43,45,47,49,50,51,53,55,57,60__,...

U digit# = r**e**a**d**.**v**e**r**t**i**c**a**l**l**y .**1 ** 1 **1**1 **1**1 **1**1 **1**1 **2**2 **2**2
**2**2 **2**2 **2**2 **3**3 **3**3
**3**3 **3**3 **3**3 **4**4 **4**4
**4**4 **4**4 **4**4 **5**5 **5**5
**5**5 **5**5 **5**5 **6**6 **6**6** ...**

U digit# = __1 2 3 4 5 6 7 8 __9

__Exam__p__le__:

for n=1, a(n)=2 whose
1^{st} digit 2 is not the #2 digit of U
-- which is a 1

for n=2, a(n)=1 whose
1^{st} digit 1 is not the #1 digit of U
-- which is a 2

for n=3, a(n)=4 whose
1^{st} digit 4 is not the #4 digit of U
-- which is a 3

...

for n=8, a(n)=7 whose
1^{st} digit 7 is not the #7 digit of U -- which is a 8

for n=9, a(n)=10
whose 1^{st} digit 1 is not the #**10**
digit of U -- which
is a 0 -- the 0 from 10

for n=10, a(n)=9
whose 1^{st} digit 9 is not the #**9**
digit of U -- which
is a 7

for n=11, a(n)=11
whose 1^{st} digit 1 is not the #**11**
digit of U -- which
is a 9

...

for n=20, a(n)=31
whose 1^{st} digit 3 is not the #**31**
digit of U -- which
is a 1 -- the 1 from 31

... so the 1^{st} digit of a(n) is never the a(n)^{th} digit of U.

We see here that
12 and 13, for instance, will never appear in U -- as 12 or 13 say that the 12^{th} or
13^{th} digit of U
_are not_ 1, which is false: the 12^{th} and the 13^{th}
digit of U are both
1 -- coming from 11.

_____

**Ray Chandler** has just checked the U sequence above and computed
100 terms; here they are:

U = 2, 1, 4, 3, 6, 5, 8,
7, 10, 9, 11, 15, 17, 19, 21, 23, 25, 27, 29, 31, 34, 40, 33, 35, 39, 41, 43,
45, 47, 49, 50, 51, 53, 55, 57, 60, 61, 63, 65, 67, 69, 70, 72, 73, 75, 77, 79,
80, 81, 82, 83, 84, 85, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100,
101, 102, 103, 104, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117,
118, 119, 120, 121, 123, 124, 126, 129, 130, 132, 133, 135, 136, 138, 139, 141,
142, 144}

_____

And well get
three more sequences like this if we replace first by last, in the
definition; this gives:

Every a(n) says: "My *last*
digit is _not_ the a(n)^{th} digit of V"

Here is a
re-ordering V of
the Naturals obeying the rule « The *last*
digit of a(n) is _not_ the a(n)^{th}
digit of V »:

-----> V = __2,1,4,3,6,5,8,7,11,9,10,12,13,14,15,16,17,18,20,19,30,21,22,23,24,31,26,25,27,28,32,29,33,34,35,36__,...

V digit# = r**e**a**d**.**v**e**r**t**i**c**a**l**l**y .**1 **1 **1**1 **1**1 **1**1
**1**1 **2**2 **2**2
**2**2 **2**2 **2**2 **3**3 **3**3
**3**3 **3**3 **3**3 **4**4 **4**4
**4**4 **4**4 **4**4 **5**5 **5**5
**5**5 **5**5 **5**5 **6**6 **6**6 **...**

V digit# = __1 2 3 4 5 6 7 8 9 0 1 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 45 67 89 01 23 ...__

The
building method is simple: we must prolong V with the smallest unused integer which has no digit
matching the corresponding digit of _the last line_ of the above array (the
line below the line with read vertically).

_____

**Ray Chandler** has checked the V sequence above and
computed 100 terms; here they are:

V = 2, 1, 4, 3, 6, 5, 8, 7, 11, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, 19,
30, 21, 22, 23, 24, 31, 26, 25, 27, 28, 32, 29, 33, 34, 35, 36, 37, 38, 40, 39,
41, 50, 42, 43, 44, 45, 51, 46, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 60, 59,
61, 62, 70, 63, 64, 65, 66, 71, 67, 68, 69, 72, 73, 74, 75, 76, 77, 78, 90, 79,
80, 81, 82, 91, 83, 84, 86, 85, 92, 87, 88, 89, 93, 94, 95, 96, 97, 98, 101,
99,...

Any taker for the last two sequences, W and X?

W: we drop the re-ordering of the Naturals but we want W to be monotonically
increasing

X: we drop both the re-ordering of the Naturals and the monotonically
increasing rule.

**Ray **has computed both sequences (100
terms); here they are:

W = 2, 3, 4, 5, 6, 7, 8,
9, 11, 20, 21, 22, 23, 24, 25, 26, 27, 29, 31, 33, 40, 41, 42, 43, 44, 45, 47,
49, 51, 53, 55, 60, 61, 62, 63, 65, 67, 69, 71, 73, 75, 77, 80, 81, 83, 85, 87,
89, 100, 101, 102, 103, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210,
211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226,
227, 28, 229, 230, 231, 232, 233, 234, 235, 236, 300, 301, 302, 304, 305, 306,
307, 400, 401, 402, 403,...

X = 2, 1, 4, 3, 6, 5, 8, 7, 11, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43,
44, 45, 46, 47, 48, 49, 50, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65,
66, 67, 68, 69, 70, 71, 72, 73, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87,
88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 101, 102, 103, 104, 105, 107,
108, 109,...

__________

**P.-S**.

Working on this
page suddenly makes me think of a nice (?) Y sequence:

« First lexically
re-ordering of the Naturals where neither a(n)^{th} first or a(n)^{th}
last digit are the a(n)^{th} digit of Y »

We just have to
combine the building methods of S and V
to get Y no digit
of Y can match any **bold** or grey digit below it:

**N**
= __1|2|3|4|5|6|7|8| 9|10|11|12|13|14|15|16|17|18|19|20|21|22|23|24|25|26|27|28|29|30|31|32|33|34|35|36|37__|...

-----> Y = __2,1,4,3,6,5,8,7,12,
9,30,20,22,23,10,11,13,14,15,16,17,18,19,21,25,31,26,28,27,24,32,29,33,34,35,37,38__,...

Y digit# = 1 2 3 4 5 6 7 8 9**1 **1 **1**1 **1**1 **1**1
**1**1 **2**2 **2**2
**2**2 **2**2 **2**2 **3**3 **3**3
**3**3 **3**3 **3**3 **4**4 **4**4
**4**4 **4**4 **4**4 **5**5 **5**5
**5**5 **5**5 **5**5 **6**6 **6**6
**6**6** ...**

Y digit# = __r ead__

_____

**Ray Chandler** has corrected my Y sequence above and computed
100 terms; here they are:

Y = 2, 1, 4, 3, 6, 5, 8, 7, 12, 9, 30, 20, 22, 23, 10, 11, 13, 14, 15, 16,
17, 18, 19, 21, 25, 31, 26, 28, 27, 24, 32, 29, 33, 34, 35, 37, 38, 39, 40, 36,
41, 50, 42, 43, 44, 45, 51, 46, 47, 48, 52, 53, 54, 55, 49, 56, 57, 58, **60**, 59, 62, 63, 70, 64, 65, 66, 67, 72,
68, 69, 74, 73, 75, 76, 77, 78, 79, 80, 90, 82, 84, 83, 85, 92, 86, 87, 88, 89,
93, 94, 95, 96, 98, 97, 99, 300, 200, 202, 204, 203,...

Ray
adds two more sequences, Y1 and Y2;

_____

This suggests the
ultimate Z sequence
(many thanks to **Franklin T.
Adams-Watters** and **William Keith**
for their interesting remarks see the Seqfan archives around February 21^{st}, 2010):

« No digit of a(n) is the a(n)^{th}
digit of Z »

**Ray Chandler** computed 100 terms which diverge from Y at Z(59)=**62** (instead of 60 see bold
figure **60** above):

Z = 2, 1, 4, 3, 6, 5, 8, 7, 12, 9, 30, 20, 22, 23, 10, 11, 13, 14, 15, 16,
17, 18, 19, 21, 25, 31, 26, 28, 27, 24, 32, 29, 33, 34, 35, 37, 38, 39, 40, 36,
41, 50, 42, 43, 44, 45, 51, 46, 47, 48, 52, 53, 54, 55, 49, 56, 57, 58, **62**, 59, 60, 63, 70, 64, 65, 66, 67, 73,
68, 69, 72, 74, 75, 76, 77, 78, 79, 80, 90, 82, 84, 83, 86, 92, 85, 87, 88, 89,
93, 94, 95, 96, 98, 202, 97, 99, 200, 203, 204, 205,...

The Z sequence is no re-ordering of the Naturals as the integer
1023456789 will never appear...

___________

Now **Ray**
adds four more sequences Y2, Y3, Z2, Z3:

>Y2, Y3 are the
variations of Y as T and U are to S.

>Z2, Z3 are the
variations of Z as T and U are to S.

Y2 = 2, 3, 4, 5, 6, 7, 8, 9, 11, 20, 21, 22, 23, 24, 30, 40, 41, 42, 44, 45,
46, 47, 48, 49, 50, 51, 60, 61, 62, 63, 64, 65, 66, 67, 70, 71, 72, 80, 81, 83,
84, 85, 86, 87, 90, 91, 92, 93, 94, 200, 201, 202, 203, 204, 205, 206, 207,
208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223,
224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 300, 301, 303,
304, 305, 306, 307, 400, 401, 410, 411, 412, 413, 414,...

Y3 = 2, 1, 4, 3, 6, 5, 8, 7, 11, 9, 14, 20, 13, 15, 17, 19, 21, 22, 23, 25,
29, 30, 31, 33, 35, 37, 40, 41, 42, 43, 45, 47, 49, 50, 51, 52, 55, 56, 57, 59,
60, 62, 63, 64, 66, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 84,
85, 86, 87, 88, 89, 90, 91, 92, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103,
104, 105, 106, 109, 110, 111, 112, 113, 114, 115, 116, 119, 120, 121, 122, 123,
124, 126, 127, 128, 129, 130,...

Z2 = 2, 3, 4, 5, 6, 7, 8, 9, 11, 20, 21, 22, 23, 24, 30, 40, 41, 42, 44, 45,
46, 47, 48, 49, 50, 51, 60, 61, 62, 63, 64, 65, 66, 67, 70, 71, 72, 80, 81, 83,
84, 85, 86, 87, 90, 91, 92, 93, 94, 200, 201, 202, 203, 204, 205, 206, 207,
208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223,
224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 311, 313, 314,
330, 333, 334, 335, 400, 401, 410, 411, 412, 413, 414,...

Z3 = 2, 1, 4, 3, 6, 5, 8, 7, 11, 9, 14, 20, 13, 15, 17, 19, 21, 22, 23, 25,
29, 30, 31, 33, 35, 37, 40, 41, 42, 43, 45, 47, 49, 50, 51, 52, 55, 56, 57, 59,
60, 62, 63, 64, 66, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 84,
85, 86, 87, 88, 89, 90, 91, 92, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103,
104, 106, 109, 110, 111, 112, 113, 114, 115, 116, 119, 120, 121, 122, 123, 124,
126, 128, 129, 130, 131, 132,...

Many thanks to all the contributors! -- Those
seq. will be submitted soon to Neils **OEIS**.

_______________________

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