[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.

[Neil Sloane]

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

On Wed, Aug 27, 2014 at 1:19 AM, Wayne VanWeerthuizen <waynemv at gmail.com> wrote:
> 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
>
>
>
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Dear Friends, I have now retired from AT&T. New coordinates:

Neil J. A. Sloane, President, OEIS Foundation
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