Pattern in A382250 Regarding A000005

[Ali Sada]

Hi everyone,

Hope all is well. 

There is an interesting pattern in A382250 regarding the number of factors of n. There are lonely vertical pairs in each sheet. These pairs count the factors of n. If the number of these pairs equals 2, then n is a prime. The pairs follow a certain pattern. Let's say the pair (a,b) with a<b, and d = b-a. and let c be the largest odd number <= a. 1+(d/c) is a power of 2. For example, let's take the pair (8,57). a = 8, b = 57, c = 7, and d = 57-8 = 49. 1+(d/c) = 1+(49/7) = 1+7 = 8.

Please see the table below for n =1 to 11. The powers are the factors of n. I don't know why this is happening, but I think it's correct for all numbers. My question is: can we conjecture a direct formula for A000005 based on this pattern? 

1            2

2            2, 4

3            2, 8

4            2, 4, 16

5            2, 32

6            2, 4, 8, 64

7            2, 128

8            2, 4, 16, 256

9            2, 8, 512

10         2, 4, 32, 1024

11         2, 2048

Best,

Ali