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A349482
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Bases where the n-th Goodstein sequence starting in base 3 (instead of base 2) reaches 0.
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0
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4, 5, 7, 9, 11, 15, 19, 23, 63, 159, 2047, 10239, 49151, 1048575, 20971519, 402653183, 1180591620717411303423, 233840261972944466912589573234605283144949206876159
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OFFSET
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1,1
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COMMENTS
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a(A056004(n)) lists the bases where the n-th Goodstein sequence starting in base 2 reaches 0. That sequence goes 3, 5, 7, 3*2^402653211 - 1, ...
The Goodstein function is sometimes given as the base where the sequence last has a nonzero value. Following this definition decreases each term in the above sequence by 1.
Like the Goodstein function (which starts in base 2), this sequence appears to grow faster than f_alpha if and only if alpha is smaller than epsilon_0.
As given by the formula below, the sequence continues with a(20,...,26) = 3*2^391 - 1, 4*2^2057 - 1, 5*2^10251 - 1, 3*2^49166 - 1, 4*2^1048594 - 1, 5*20971540 - 1, 3*402653211 - 1.
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LINKS
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FORMULA
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For n<27, i.e. n = a*3^2 + b*3 + c with a, b, c < 3, a(n) = f_2^a (f_1^b (f_0^c (4) ) ) - 1.
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EXAMPLE
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For n=6, we write 6 = 2*3.
Then in each step, we increase the base by 1, then subtract 1. This sequence goes
2*3 -> 1*4 + 3 -> 1*5 + 2 -> 1*6 + 1 -> 1*7 + 0 -> 0*8 + 7 ->...-> 0*14 + 1 -> 0*15, so 0 is reached at base 15.
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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STATUS
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approved
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