A137389

[Alexander Povolotsky]

Dear Neil

 I noticed that you have added terms to original version of A137389

  [12, 40, 160, 2176, 10240, 139264, 655360, 8912896, 545259520,
  2684354560, 139586437120, 2336462209024, 10995116277760,
  149533581377536, 9147936743096320, 585467951558164480,
  2882303761517117440, 149879795598890106880, 2508757194024499019776,
  11805916207174113034240 ]

1,1
 and corrected the error (which I pointed to you) in the A137389's
  example.

 From the A137389's definition

a(n) = 2^(prime(n)) + 2^(prime(n + 1))

it is obvious that all terms in A137389 are divided by
2^(prime(1))=2^2=4

So it would make sense to revise this sequence with above suggested reduction
(division by 4)

a(n) = (2^(prime(n)) + 2^(prime(n + 1))) / 2^(prime(1))

 [3, 10, 40, 544, 2560, 34816, 163840, 2228224, 136314880, 671088640,
34896609280, 584115552256,  2748779069440, 37383395344384,
2286984185774080, 146366987889541120, 720575940379279360,
37469948899722526720, 627189298506124754944, 2951479051793528258560]

Going further ... additionally (or instead of A137389) it would make
sense to have the sequence, defined as follows:

a(n) = 1 + 2^( prime(n + 1) - prime(n) )

[3, 5, 5, 17, 5, 17, 5, 17, 65, 5, 65, 17, 5, 17, 65, 65, 5, 65, 17,
5, 65, 17, 65, 257, 17, 5, 17, 5, 17, 16385, ...]

Thanks,
Best Regards,
Alexander R. Povolotsky
pevnev at juno.com