[seqfan] Re: A027763 and related sequences regarding smallest k such that b^^n is not congruent to b^^(n-1) mod k, for constant b.

[Wayne VanWeerthuizen]

Thanks for the feedback, Neil.

I went ahead and submitted sequences for bases 3 through 9 as A246491 through A246497. The first I proposed for review. The others I've left as drafts until I get a feel for what changes the editors deem need to be made to A246491.

I submitted my source code for A246491 as a linked file; I hope the manner in which I did so is okay. I'd like to link to the same file for the related sequences, if that is okay.

For A246497, with base 9, the server warned me that it may be the same as A082449. So we now have two pairs of sequences to check for equivalence:
  Is A027763 essentially the same as A173927?
  Is A246497 essentially the same as A082449?

I've made further discoveries while examining the initial terms of the sequences
for much higher bases:

  3 occurs as a term in sequences with
     base=2, 3, 5, 8, 9, 11, 14, 15, 17, 20, 21, 23, 26, 27, 29, 32, 33, ...

 25 occurs as a term in sequences with
     base=22, 67, 78, 87, 98, 122, 142, 144,187,198,208,242,287,298,342

125 occurs as a term in sequences with
    base=342, 847, 1408, 2287, 2662, 2838, 4302, 6478, 7008, 8238,...

 49 occurs as a term in sequences with
    base=639, 737, 4436, 5841, 7569, 8306,...

121, 169, 289, 343, and 361 were not found as a term in any base < 10000.

Lowest base in which each prime occurs, list of (prime,base)
  (2,2), (3,2), (5,2), (7,5), (11,2), (13,6), (17,10), (19,5), (23,2),
  (29,26), (31, 43), (37,69), (41,220), (47,2), (53,34), (59,11), ...

So, possibly all primes eventually occur for some b.
I am less sure whether all powers of primes occur, given that 121, 169, 289, 343, etc. could not easily be found.

Would any of the meta-sequences mentioned in this post themselves make good candidates for being submitted to the database, or are they too abstract/obscure and difficult to describe in a general enough way?

Wayne VanWeerthuizen


On 8/27/2014 5:16 AM, Neil Sloane wrote:
> That's a very interesting question, is A027763 essentially the same as A173927!
> 
> Certainly you should submit the analogs for bases 3 through 7.
> 
> Neil
>