A121760/1: two (interesting?) sequences

[Paul D. Hanna]

Seqfans, 
     In general, I agree with Joerg:
> When 'base' meets 'prime' all hope is lost.
> When 'prime' meets addition, all hope is lost.
> When 'decimal' meets about anything, all hope is lost.
 
but of course, there are exceptions. 
 
A notable exception to base sequences are the 10-adic integers; 
below I give some examples of 10-adic integers in the OEIS. 
And other interesting base-related sequences do exist. 
 
And for primes, I've always thought they were overused, 
yet I submitted A099863 (below) because the distribution 
of the primes modulo 2^n arouses my curiousity 
(due in part to Wallis' approximation to Pi/2). 
 
There are better examples of valuable prime-related sequences, 
but I am giving A099863 a shameless plug.  : )
 
My point is, don't be too quick to "throw out the baby with the bath
water". 
But I agree that such sequences can be trivial clutter 
for obvious reasons that have been clearly stated. 
 
Paul
 
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A099863  
A permutation of the odd primes that satisfy the condition that 
the sequence modulo 2^n has period 2^(n-1) for all n>0, where 
the least unused primes are chosen in the process.  

The sequence is of period 2^(n-1) modulo 2^n, for all n>0, 
and consists of all odd numbers less than 2^n: 
[1, 1, 1, 1, 1, 1, 1, 1, 1, ...] (mod 2) 
[3,1, 3,1, 3,1, 3,1, 3,1, 3,1, ...] (mod 4) 
[3,5,7,1, 3,5,7,1, 3,5,7,1, 3,5,7,1, ...] (mod 8) 
[3,5,7,1,11,13,15,9, 3,5,7,1,11,13,15,9, ...] (mod 16) 
[3,5,7,17,11,13,31,9,19,21,23,1,27,29,15,25, ...] (mod 32)
  
First 256 terms: primes that remain primes (mod 2^9) 
are indicated by 'O', composites indicated by '.' 
(modulo 2^9, the sequence A099863 is congruent to 
all odd numbers < 2^9): 
 
OOOOOOOOOOOOOOOO 
OOOOOOOOOOOOOOOO 
OOOOOOOOOO..OOOO 
OOOOOOOOO.OOOOO. 
OO.OOO...OO.OOO. 
.OOO...OOOO...O. 
..O.....O...O.OO 
.O..O........... 
..O..O.....O.... 
OOO...O....O.O.. 
......O......... 
................ 
................ 
................ 
........O....... 
................ 
 
 
First 1024 terms: primes that remain primes (mod 2^11) 
are indicated by 'O', composites indicated by '.' 
(modulo 2^11, the sequence A099863 is congruent to 
all odd numbers < 2^11): 
 
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 
OOOOOOOOOOOOOOOOOOOOOOOOOOOO.OOO 
OOOOOOOOOOO.OOOOOOO.OOOOOOOOO.OO 
OOOOOOOOOOOOOOOOOOOOOOO.OO.OOOOO 
..OOOOO...OOOO.O.OOO.OO..OO.OOO. 
.O..OO.OOOOOOO...OOOO...O.....OO 
.OO...O.OOO......OO.O..O.OOO...O 
..OOOOO...O.OOOOO...O.OOOOO..O.. 
.OOOOOO.O...O.O.OO..OO....OOOO.. 
O....OO.O...OO.....O...O.....OO. 
...OO..OO...........OO.O...OO.O. 
.......O...O............O....O.. 
.....OO...O.O........O...O...O.. 
..............................O. 
........O.........O....O..O..... 
..O.............OO....OO...OO.O. 
O..O.......O.O..........OO...... 
....O........O..O.O....O.O.O..O. 
..O............O........O....... 
........O............O.......... 
................................ 
....O........................... 
................................ 
..........................O..... 
.O...............O.............. 
O............................... 
................................ 
................................ 
................................ 
................................ 
................................ 
 
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A018247  
The 10-adic integer x = ...8212890625 satisfies x^2 = x.  
 
x = 10-adic Limit_{n->infty} 5^(2^n) mod 10^(n+1). 
   
A120817
10-adic integer x=...07839804103263499879186432 satisfying x^5 = x; 
also x^3 = -x = A120818; 
(x^2)^3 = x^2 = A091664; 
(x^4)^2 = x^4 = A018248. 
 
x = 10-adic Limit_{n->infty} 2^(5^n) mod 10^(n+1). 
 
x^1=...3304553032451441224165530407839804103263499879186432 (A120817). 
x^2=...0557423423230896109004106619977392256259918212890624 (A091664). 
x^3=...6695446967548558775834469592160195896736500120813568 (A120818). 
x^4=...9442576576769103890995893380022607743740081787109376 (A018248). 
x^5=...3304553032451441224165530407839804103263499879186432 = x. 
 
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