The On-Line Encyclopedia of Integer Sequences (OEIS)

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A246491

Smallest k such that 3^^n is not congruent to 3^^(n-1) mod k, where 3^^n denotes the power tower 3^3^...^3 (in which 3 appears n times).
(history; published version)

#32 by R. J. Mathar at Fri Aug 02 05:00:48 EDT 2024

STATUS

editing

approved

#31 by R. J. Mathar at Fri Aug 02 05:00:40 EDT 2024

CROSSREFS

Large overlap with A056637.

STATUS

approved

editing

#30 by N. J. A. Sloane at Sat Nov 29 12:46:49 EST 2014

STATUS

editing

approved

#29 by Wayne VanWeerthuizen at Fri Sep 12 21:04:35 EDT 2014

COMMENTS

This sequence is a member of an interesting class of sequences defined by the rule, "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 some constant b. Different choices for b, with b>=2 , determine a sequence of this class. The first sequence of this class is A027763, which uses b=2. This is the sequence for b=3. Adjacent sequences follow for b=4 through b=9.

It is easily proven that sequences of this class can contain only terms that are either prime numbers or powers of primes (A000961). (In the general case, given any natural numbers x and y, the least k, such that x is not congruent y mod k, must be a prime or prime power.)

Sequences of this class can contain only terms that are either prime numbers or powers of primes (A000961).

For successive values of b, the sequence of first terms of all sequences of this class is A007978, which contains all and only the prime powers p^k, k>0.

It appears that the sequence for b=x contains the term y, if and only if the sequence for b=x+A003418(y) also contains the term y, where A003418(y) is the least common multiple of all the integers from 1 to y. Can this be proved?

Sequences of this class generally share many terms with other sequences of this class, but it although each appears that each sequence of this class is still to be unique. The These individual sequences are very similar to each other, so much so that among the first 7000 sequences of this class, them, only 120 distinct values less than 50000 occur (compare that to the 5217 available primes or powers of primes that are less than 50000.)

Discussion

Sun Nov 02

12:06

OEIS Server: This sequence has not been edited or commented on for a week
yet is not proposed for review.  If it is ready for review, please
visit https://oeis.org/draft/A246491 and click the button that reads
"These changes are ready for review by an OEIS Editor."

Thanks.
  - The OEIS Server

Tue Nov 04

12:42

Joerg Arndt: Could you please work on this edit?

Wed Nov 26

05:24

OEIS Server: This sequence has not been edited or commented on for a week
yet is not proposed for review.  If it is ready for review, please
visit https://oeis.org/draft/A246491 and click the button that reads
"These changes are ready for review by an OEIS Editor."

Thanks.
  - The OEIS Server

#28 by Wayne VanWeerthuizen at Fri Sep 12 20:17:12 EDT 2014

COMMENTS

This sequence is a member of an interesting class of sequences defined by the rule, "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 some constant b. Different choices for b, with b>=2 determine a sequence of this class. The first sequence of this class is A027763, which uses b=2. This is the sequence for b=3. Adjacent sequences follow for b=4 through b=9.

Different choices for b, with b>=2 determine a sequence It is easily proven that sequences of this class can contain only terms that are either prime numbers or powers of primes (A000961). (In the general case, given any natural numbers x and y, the least k, such that x is not congruent y mod k, must be a prime or prime power.)

The first sequence Powers of three (A000244) are commonplace as terms in sequences of this class is A027763, which uses b=2, occurring significantly more often than powers of other primes.

This is For successive values of b, the sequence for b=3. Adjacent of first terms of all sequences follow for b=4 through b=11of this class is A007978, which contains all and only the prime powers p^k, k>0.

It appears that sequences of this class may contain only terms that are either prime numbers or powers of primes (A000961).

Powers of three (A000244) are commonplace as terms in sequences of this class, occurring significantly more often than powers of other primes, and about as often as primes.

All primes up to 179 have been confirmed to occur among sequences of this class. 181 doesn't occur for b<13500000. The term 113 wasn't found until b=10337120.

For the particular powers of primes that have been spotted so far among sequences of this class, here are the lowest b values for which they occur: 2^2=4 at b=6, 3^2=9 at b=14, 5^2=25 at b=22, 2^5=32 at b=28981, 7^2=49 at b=639, 2^6=64 at b=1611, 11^2=121 at b=63687, 5^3=125 at b=342, 2^7=128 at b=38501, 2^8=256 at b=451, 7^3=343 at b=2461706, 5^4=625 at b=2542. Neither 169, 289, 361, nor 529 have been spotted for b<13000000, but it's not an unreasonable conjecture that for any prime or power of a prime there is a sequence of this class in which it occurs.

STATUS

approved

editing

#27 by N. J. A. Sloane at Sat Sep 06 00:19:20 EDT 2014

STATUS

editing

approved

#26 by N. J. A. Sloane at Sat Sep 06 00:19:16 EDT 2014

COMMENTS

This sequence is a member of an interesting class of sequences following defined by the rule, "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 some constant b.

STATUS

proposed

editing

#25 by Wayne VanWeerthuizen at Tue Sep 02 13:47:38 EDT 2014

STATUS

editing

proposed

#24 by Wayne VanWeerthuizen at Tue Sep 02 13:47:06 EDT 2014

LINKS

Wayne VanWeerthuizen, <a href="/A246491/a246491_34.txt">Initial terms of sequences of this class for b=2..100010001</a>

STATUS

proposed

editing

#23 by Wayne VanWeerthuizen at Tue Sep 02 05:01:57 EDT 2014

STATUS

editing

proposed

Discussion

Tue Sep 02

05:54

Michel Marcus: Not familiar with sage
but if it has powermod and eulerphi why rewrite your own ?
then code could be in program section ?

07:09

Wayne VanWeerthuizen: At first, I rewrote the functions because I was having weird performance issues with the built-in functions and I didn't know the cause. (Sage kept crashing on me due to a memory leak.) I later realized that the leak only happened when I used Sage's notebook feature, but I kept using my own functions anyway once I switched to using Sage just from the command line. My functions may be just as efficient (if not better than) the built-in ones if Cython compiles them well, as they use strong data typing and don't require any type conversions. I should probably run some actual benchmarks to find out for sure. But another reason the code is not in the program section is that by making it a linked file, I can link to the same code from A246492 thru A246497. Also, I think that even if I used the built in power_mod and euler_phi functions, the code would still be too long for the program section, given how many lines my tower_mod function is just by itself.  Furthermore, I was being optimistic that someone else would contribute good Mathematica or Maple code, so that the Sage code would soon be redundant anyway.