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A321105
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One of the three successive approximations up to 13^n for 13-adic integer 5^(1/3). This is the 11 (mod 13) case (except for n = 0).
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12
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0, 11, 154, 999, 25166, 82288, 82288, 43523569, 43523569, 4937907895, 121587400998, 1362313827639, 12115276191861, 175201872049228, 2901077831379505, 10775830602778083, 471448867729594896, 6460198350378213465, 23761030189140889331, 361127251045013068718, 4746888122171351400749
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OFFSET
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0,2
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COMMENTS
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For n > 0, a(n) is the unique number k in [1, 13^n] and congruent to 11 mod 13 such that k^3 - 5 is divisible by 13^n.
For k not divisible by 13, k is a cube in 13-adic field if and only if k == 1, 5, 8, 12 (mod 13). If k is a cube in 13-adic field, then k has exactly three cubic roots.
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LINKS
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EXAMPLE
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The unique number k in [1, 13^2] and congruent to 11 modulo 13 such that k^3 - 5 is divisible by 13^2 is k = 154, so a(2) = 154.
The unique number k in [1, 13^3] and congruent to 11 modulo 13 such that k^3 - 5 is divisible by 13^3 is k = 999, so a(3) = 999.
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PROG
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(PARI) a(n) = lift(sqrtn(5+O(13^n), 3) * (-1-sqrt(-3+O(13^n)))/2)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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