[seqfan] 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]

I was recently examining sequence A027763, since it came up as a related series while I was working on A245970 and A240162.

A027763 is "Smallest k such that 2^^n is not congruent to 2^^(n-1) mod k, where 2^^n denotes the power tower 2^2^...^2 (in which 2 appears n times)." For the sake of this discussion, I'll call A027763 the b=2 sequence.

I was wondering what would happen if the 2 was replaced with other values to get a new sequences. The b=3 sequence would give the "smallest k such that b^^n is not congruent to b^^(n-1) mod k, where b=3 and b^^n denotes the power tower b^b^...^b (in which b appears n times)."

After running many calculations, what fascinates me is how dramatically similar these sequences are to each other, in spite occasionally differing.

At the bottom of this post are all the terms I've been able to calculate so far. All the terms encountered so far are either prime, or are a power of a prime.

Many of the same terms occur frequently in different sequences in this list of sequences.
   23 occurs in b=2,3,4,7,8,9,13,17,18,19,...
   47 occurs in b=2,3,4,5,6,7,8,9,10,13,14,15,16,17,18,19,...
  283 occurs in b=2,3,4,7,8,9,10,13,15,17,18,...
  719 occurs in b=2,3,4,5,7,8,9,10,13,15,17,18,19,...
 1439 occurs in b=2,3,4,5,6,7,8,9,10,12,13,15,17,18,19,21,...

Does every prime eventually occur in at least one of these sequences? Lowest primes not yet seen: 29, 31, 37, 41, 53, ...
If not, is there a rule for which primes can or cannot occur?
Once an odd term occurs, are all subsequent terms also odd? (Proving this would allow for significant optimization of the calculation of new terms.)
Does every power of three eventually occur? Powers of three not yet seen: 3^5, 3^6, 3^7, 3^17, ...
Does every prime-squared occur?  Seen so far: 2^2 (at b=6), 3^2 (at b=14), 5^2 (at b=22),
Does two-cubed or five-cubed ever occur?

Also, can it be proved that the b=2 sequence (A027763) is the same as A173927, based on iterations of Carmichael lambda function, except for the initial term?

Does anyone have any further insights or interesting conjectures about these sequences?

How many of them should I add to the OEIS? I am thinking I should add b=3 through b=7, does that sound good?

Terms I've calculated:
  (Terms are in brackets are perfect powers that have been factored. Otherwise the terms are all prime.)

b= 2: 2, 3, 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, [3^12], [3^13], [3^14], [3^15], [3^16], 86093443, 344373773, ...

b= 3: 3, 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, 14920303, 36449279, ...

b= 4: 2, 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, [3^14], [3^15], ...

b= 5: 3, 7, 19, 47, 243, 719, 1439, 2879, [3^9], [3^10], [3^11], [3^12], [3^13], [3^14], ...

b= 6: 2, [2^2], 13, 47, 107, 643, 1439, 2879, 34549, 138197, 858239, 2029439, 36449279, ...

b= 7: [2^2], 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, [3^14], [3^15], ...

b= 8: 2, 3, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, ...

b= 9: 3, 7, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, 14619833, ...

b=10: 2, [2^2], 17, 47, 283, 719, 1439, 2879, 34549, 138197, 858239, 3778253, ...

b=11: 3, 7, 17, 59, 163, 487, 1307, [3^8], [3^9], [3^10], [3^11], [3^12], 1594323, 4782969, 14348907, ...

b=12: 2, 5, 17, 83, 167, 503, 1439, 2879, 34549, 138197, 858239, ...

b=13: 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, [3^12], [3^13], [3^14], [3^15], ...

b=14: 3, [3^2], 19, 47, 163, 487, 1307, 2879, [3^9], 39367, [3^11], [3^12], ...

b=15: 3, [3^2], 19, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, 8826479, ...

b=16: 2, 17, 47, 163, 487, 1307,  2879, [3^9], [3^10], [3^11], [3^12], 1594323, ...

b=17: 3, 7, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, ...

b=18: 2, [2^2], 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, ...

b=19: [2^2], 11, 23, 47, 257, 719, 1439, 2879, 34549, 138197, 1266767, ...

b=20: 2, 3, [3^2], [3^3], [3^4], 163, 487, 1307, [3^8], [3^9], [3^10], [3^11], [3^12], ...

b=21: 3, 9, 19, 149, 383, 1439, 2879, 32633, 65267, 913739, 1827479, ...

b=22: 2, [2^2], 5, [2^5], 83, 167, 503, 3019, 14087, 84523, 482957, 1449167, 4782969, ...

-- Wayne VanWeerthuizen