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A246491
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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).
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8
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3, 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, 14920303, 36449279, 377982107
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OFFSET
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1,1
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COMMENTS
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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.
Sequences of this class can 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.
For successive values of b, the sequence of first terms of all sequences of this class is A007978.
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, although each appears to be unique. These individual sequences are very similar to each other, so much so that among the first 7000 of 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.)
Sequences of this class tend to share many terms with A056637, the least prime of class n-.
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LINKS
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EXAMPLE
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(3^^3) mod 11 = 9, (3^^2) mod 11 = 5. 11 is the least whole number for which this is true, thus a(3)=11.
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PROG
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(Sage, cython) Use linked source above with this command: find_k_values(3, 1, 1)
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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