[seqfan] Producing primes from powers of 2

[Ali Sada]

Hi Everyone,

We want to produce prime numbers out of “blocks” of powers of 2. Each time we use one block in addition to the remains of the previous block. When we start using a block of size 2^k, we cannot go back later and use smaller blocks.

To get 2, we need a block of 2. So, a(1) = 2-2 = 0 

a(1) = 0 

To get 3, we need a block of 4. So, a(2) = 4-3 = 1

a(2) = 1

To get 5, we need a block of 4 in addition to the remains of the previous block. a(3) = 4+1-5 = 0.

a(3) = 0

We continue:

a(4) = 8-7 = 1

a(5) = 16+1-11= 6

a(6) = 16+6-13 = 9

a(7) = 16+9-17= 8

a(8) = 16+8-19 = 5

a(9) = 32+5-23 = 14

and so on.

This is the sequence we get:

0, 1, 0, 1, 6, 9, 8, 5, 14, 17, 18, 13, 4, 25, 42, 53, 58, 61, 58, 51, 42, 27, 8, 47, 78, 105, 130, 151, 170, 185, 186, 183, 174, 163, 142, 119, 90, 55, 16, 99, 176, 251, 316, 379, 438, 495, 540, 573, 602, 629, 652, 669, 684, 689, 688, 681, 668, 653, 632, 607, 580, 543, 492, 437, 380, 319, 244, 163, 72

 

I calculated 10,000 terms, and the graph seems like a “smooth” quasiperiodic function.  


I would really appreciate your help defining this sequence.


Best,

 
Ali