[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.

Tue Sep 2 12:51:54 CEST 2014

I was hoping to have more discussion of A027763, and A246491-A246497.  There are still open questions about the sequences and details I could use help with.  Also, I have also noticed two instances where sequences of this class appear to coincide with other sequences in the OEIS database. Finding out more about these sequences as a class may shed more light on those other sequences.

To quickly review, these are a class of related sequences based on the modular arithmetic properties of power towers. Each sequence gives "the smallest k such that b^^n is not congruent to b^^(n-1)  mod k, where b^^n denotes the power tower b^b^...^b (in which b appears n times)," for a particular constant b. Note that throughout this post, b is always used to refer to the base (for the power tower) of a given sequence of the class, and Seq#b to the sequence of this class that uses that base. For background, please read my comments on the current draft of A246491 (Seq#3). There are also some earlier posts in this thread which bring up other questions and observations not repeated in this post.

I recently uploaded a file with the initial terms for the first 1000 sequences in this class of sequences to:

     https://oeis.org/A246491/a246491_3.txt

*** Question: Is there a formal proof that all the terms in these sequences must be primes or powers or primes?

Moving on, I've recently noticed a fascinating, regular pattern to the occurrence of terms among these sequences.

It appears that whenever b is even, the first term of Seq#b is always 2. Conversely, the first term is never 2 when b is odd.

Furthermore, the pattern appears to continue for other terms:

* Seq#b includes 3, if and only if b%6 is in {2,3,5}.

* Seq#b includes 4, if and only if b%12 is in {6,7,10}.

* Seq#b includes 5, if and only if b%60 is in {2,3,4,7,12,13,18,22,23,24,25,27,28,37,38,42,43,47,48,49,52,58}

* Seq#b includes 7 if and only if b%420 is in {5,9,11,17,26,32,33,39,40,44,45,51,53,59,60,61,65,68,74,75,76,80,81,86,89,93,95,96,101,110,116,117,121,128,129,131,135,136,137,149,152,156,159,160,164,165,170,171,173,177,179,180,181,185,191,194,200,201,206,212,213,215,216,219,220,221,233,236,240,241,248,489,254,255,257,261,269,275,276,284,285,290,291,296,297,299,300,301,305,311,317,320,326,332,333,339,341,345,353,359,360,361,368,369,374,375,376,380,381,389,395,396,401,404,410,411,416,417}.

* Seq#b includes 8 if and only if b%840 is in {36,100,196,316,340,420,421,436,540,676,700,756,820}

* Seq#b includes 9 if and only if b%2520 is in a set of 299 possible values, too many to list here.

It appears that sequences of this class never include terms that are the products of distinct primes.
To make those cases also fit this pattern, 6 and 10 might follow these rules, where the given sets are empty:

* Seq#b includes 6, if and only if b%60 is in {}.

* Seq#b includes 10 if and only if b%2520 is in {}.

A generalized rule that I have observed (but not theoretically proven):

* A term with value n occurs in seq#b, if and only if it occurs in seq#(b+LCM(1..n)).

Note that LCM(1..n) is A003418(n), the least common multiple of all k with 1<=k<=n.

I could use some help with trying to extend the generalized rule to:

* Sequence #b includes a term of value n if and only if b%A003418(n) is in the set S, with S being ....

But so far, I haven't discovered a general rule for constructing these sets.

*** Question: Can anyone help me find a general rule for determining what elements are in these sets?

Note also, the sizes of the above sets are:
  for b= 2,   1 element.
  for b= 3,   3 elements.
  for b= 4,   3 elements.
  for b= 5,  22 elements.
  for b= 6,   0 elements.
  for b= 7, 118 elements.
  for b= 8,  13 elements.
  for b= 9, 299 elements.
  for b=10,   0 elements.
etc.

I tried asking Superseeker about "1 3 3 22 0 118 13 299", but it didn't come up with anything.

*** Question: Can anyone find a rule to describe the number of elements in each of these sets?

Thanks for any help, further observations, or general feedback I can get on these problems.

Wayne