The same sequence, but differently

We will look today at sequences whose terms can be modified without this modification affecting the sequence itself.

It seems paradoxical, but look instead:

a) Add 9 to each digit of A, then follow the result with a comma; A does not change:

A = 10, 9, 18, 10, 17, 10, 9, 10, 16, 10, 9, 18, 10, 9, 10, 15, 10, 9, 18, 10, 17, 10, 9, 18, 10, 9, 10, 14, 10, 9, 18, 10, 17,...

Check:

– the 1st digit of A is 1

– we add 9 to 1 and get 10 [which is indeed a(1)]

– the 2nd digit of A is 0

– we add 9 to 0 and get 9 [which is indeed a(2)]

– the 3rd digit of A is 9

– we add 9 to 9 and get 18 [which is indeed a(3)], etc.

b) Add 18 to each digit of B, then follow the result with a comma; B stays the same:

B = 20, 18, 19, 26, 19, 27, 20, 24, 19, 27, 20, 25, 20, 18, 20, 22, 19, 27, 20,...

c) Add 27 to each digit of C, then follow the result with a comma; C does not change:

C = 30, 27, 29, 34, 29, 36, 30, 31, 29, 36,...

d) Add 36 to each digit of D, then follow the result with a comma; D stays the same:

D = 40, 36, 39, 42, ...

Etc.

What about multiplying instead od adding? Multiplying each digit by 10 or 11 is a bit dull – using 12 is more interesting:

M = 12, 24, 24, 48, 24, 48, 48, 96, 24, 48, 48, 96, 48, 96, 108, 72, 24, 48, 48, 96, 48, 96, 108, 72, 48, 96, 108, 72, 12, 0, 96, 84, 24, 24, 48, ...

Check:

– the 1st digit of M is 1

– we multiply 1 by 12 and get 12 [which is indeed m(1)]

– the 2nd digit of M is 2

– we multiply 2 by 12 and get 24 [which is indeed m(2)]

– the 3rd digit of M is 2

– we multiply 2 by 12 and get 24 [which is indeed m(3)], etc.

___________________

Next day update (Jan 25th 2024)

Giorgos Kalogeropoulos almost immediately corrected and extended the sequence A, with a commentary:

> Here are the first 100 terms

A = 10, 9, 18, 10, 17, 10, 9, 10, 16, 10, 9, 18, 10, 9, 10, 15, 10, 9, 18, 10, 17, 10, 9, 18, 10, 9, 10, 14, 10, 9, 18, 10, 17, 10, 9, 10, 16, 10, 9, 18, 10, 17, 10, 9, 18, 10, 9, 10, 13, 10, 9, 18, 10, 17, 10, 9, 10, 16, 10, 9, 18, 10, 9, 10, 15, 10, 9, 18, 10, 17, 10, 9, 10, 16, 10, 9, 18, 10, 17, 10, 9, 18, 10, 9, 10, 12, 10, 9, 18, 10, 17, 10, 9, 10, 16, 10, 9, 18, 10, 9, ...

> The only integers that appear in the sequence are [9, 10, 11, 12, 13, 14, 15, 16, 17, 18].

Here are the number of times each of those numbers appears in the first 100000 terms:

{10, 43146},

{9, 24597},

{18, 14022},

{17, 7996},

{16, 4559},

{15, 2598},

{14, 1481},

{13, 844},

{12, 482},

{11, 275}

> The index (positions) of these numbers in the seq are:

9 ~~> 2, 7, 11, 14, 18, 23, 26, 30, 35, 39, 44, 47, 51, 56, 60, 63, 67, 72, 76, 81 ... OEIS A287107

10 ~~> 1, 4, 6, 8, 10, 13, 15, 17, 20, 22, 25, 27, 29, 32, 34, 36, 38, 41, 43, 46 ... OEIS A287106

11 ~~> 151, 616, 966, 1231, 1580, 2045, 2310, 2658, 3123, 3473, 3938, 4203, 4549, 5014, 5364, 5629, 5978, 6443, 6793, 7258 ... not in OEIS

12 ~~> 86, 351, 551, 702, 901, 1166, 1317, 1515, 1780, 1980, 2245, 2396, 2593, 2858, 3058, 3209, 3408, 3673, 3873, 4138 ... not in OEIS

13 ~~> 49, 200, 314, 400, 514, 665, 751, 864, 1015, 1129, 1280, 1366, 1478, 1629, 1743, 1829, 1943, 2094, 2208, 2359 ... not in OEIS

14 ~~> 28, 114, 179, 228, 293, 379, 428, 493, 579, 644, 730, 779, 843, 929, 994, 1043, 1108, 1194, 1259, 1345 ... not in OEIS

15 ~~> 16, 65, 102, 130, 167, 216, 244, 281, 330, 367, 416, 444, 481, 530, 567, 595, 632, 681, 718, 767 ... not in OEIS

16 ~~> 9, 37, 58, 74, 95, 123, 139, 160, 188, 209, 237, 253, 274, 302, 323, 339, 360, 388, 409, 437 ... not in OEIS

17 ~~> 5, 21, 33, 42, 54, 70, 79, 91, 107, 119, 135, 144, 156, 172, 184, 193, 205, 221, 233, 249 ... not in OEIS

18 ~~> 3, 12, 19, 24, 31, 40, 45, 52, 61, 68, 77, 82, 89, 98, 105, 110, 117, 126, 133, 142 ... not in OEIS

> As it is difficult to search for patterns in this seq, here is a visualization of the first 10000 terms cut in 100 chunks of 100 terms:

> As we can see, although we can find repeating subsequences of any length, the whole pattern is not repeating.

> When we "add x to every digit" we can have at most 10 results and no other, namely x+0, x+1,... x+9

> So, in order to get more results we can use a new rule like:

Instead of having a fixed x, for every term a(n) of the seq we can add a(n) to every digit of a(n).

Starting with a(1)=1 the sequence goes like:

{1},

{2},

{4},

{8}, next step 8+8=16

{16}, next step 16+1=17 and 16+6=22

{17, 22}, next step 17+1=18, 17+7=24, 22+2=24, 22+2=24

{18, 24, 24, 24},

{19, 26, 26, 28, 26, 28, 26, 28},

{20, 28, 28, 32, 28, 32, 30, 36, 28, 32, 30, 36, 28, 32, 30, 36},

{22, 20, 30, 36, 30, 36, 35, 34, 30, 36, 35, 34, 33, 30, 39, 42, 30, 36, 35, 34, 33, 30, 39, 42, 30, 36, 35, 34, 33, 30, 39, 42},

{24, 24, 22, 20, 33, 30, 39, 42, 33, 30, 39, 42, 38, 40, 37, 38, 33, 30, 39, 42, 38, 40, 37, 38, 36, 36, 33, 30, 42, 48, 46, 44, 33, 30, 39, 42, 38, 40, 37, 38, 36, 36, 33, 30, 42, 48, 46, 44, 33, 30, 39, 42, 38, 40, 37, 38, 36, 36, 33, 30, 42, 48, 46, 44}

> The final sequence is

F = 1, 2, 4, 8, 16, 17, 22, 18, 24, 24, 24, 19, 26, 26, 28, 26, 28, 26, 28, 20, 28, 28, 32, 28, 32, 30, 36, 28, 32, 30, 36, 28, 32, 30, 36, 22, 20, 30, 36, 30, 36, 35, 34, 30, 36, 35, 34, 33, 30, 39, 42, 30, 36, 35, 34, 33, 30, 39, 42, 30, 36, 35, 34, 33, 30, 39, 42, 24, 24, 22, 20, 33, 30, 39, 42, 33, 30, 39, 42, 38, 40, 37, 38, 33, 30, 39, 42, 38, 40, 37, 38, 36, 36, 33, 30, 42, 48, 46, 44, 33, 30, 39, 42, 38, 40, 37, 38, 36, 36, 33, 30, 42, 48, 46, 44, 33, 30, 39, 42, 38, 40, 37, 38, 36, 36, 33, 30, 42, 48, 46, 44, ...

... which is not in OEIS

> F has also a fractal plot, first 4000 terms hereunder:

> First 133000 terms:

ÉA

This is just brilliant, Giorgos – many thanks!

____________________

Jan. 27th update

ÉA to GK

> Hi Giorgos (...) you explained to me that the only integers that appear in the seq A opening this page are [9, 10, 11, 12, 13, 14, 15, 16, 17 and 18]. This made me think: what about the seq T and (another/different) new rule:

« Add 9 to the 1st digit of T, 10 to the 2nd digit of T, 11 to the 3rd digit, 12 to the 4th, 13 to the 5th, etc. so that the seq T will be unchanged ».

T = 10, 10, 12, 12, 14, 16, 16, 18, 18, 22, 20, 26, 22, 28, 24, 32, 26, 34, 29, ... »

> I guess that the number of distinct integers is not limited anymore to 10 here — what do you think? does the above T work?

> Yes, it works; the first 100 terms are:

T = 10, 10, 12, 12, 14, 16, 16, 18, 18, 22, 20, 26, 22, 28, 24, 32, 26, 34, 29, 30, 31, 30, 33, 38, 35, 36, 37, 44, 39, 42, 42, 42, 43, 48, 46, 48, 47, 55, 50, 48, 52, 51, 54, 52, 56, 57, 58, 64, 60, 63, 62, 66, 64, 69, 67, 68, 68, 75, 71, 70, 73, 72, 75, 74, 77, 77, 79, 84, 81, 84, 83, 88, 85, 89, 88, 89, 90, 86, 91, 96, 94, 92, 96, 93, 98, 98, 100, 98, 102, 104, 104, 107, 106, 110, 109, 108, 111, 106, 113, 111 > Here is a 100 terms plot: