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A325892
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The successive approximations up to 2^n for the 2-adic integer 3^(1/5).
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4
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0, 1, 3, 3, 3, 19, 19, 83, 211, 211, 211, 211, 2259, 6355, 14547, 30931, 63699, 129235, 129235, 129235, 129235, 129235, 129235, 4323539, 4323539, 4323539, 4323539, 4323539, 4323539, 4323539, 4323539, 4323539, 2151807187, 6446774483, 6446774483, 6446774483
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OFFSET
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0,3
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COMMENTS
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a(n) is the unique solution to x^5 == 3 (mod 2^n) in the range [0, 2^n - 1].
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LINKS
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FORMULA
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For n > 0, a(n) = a(n-1) if a(n-1)^5 - 3 is divisible by 2^n, otherwise a(n-1) + 2^(n-1).
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EXAMPLE
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For n = 2, the unique solution to x^5 == 3 (mod 4) in the range [0, 3] is x = 3, so a(2) = 3.
a(2)^5 - 3 = 240 which is divisible by 8, so a(3) = a(2) = 3;
a(3)^5 - 3 = 240 which is divisible by 16, so a(4) = a(3) = 3;
a(4)^5 - 3 = 240 which is not divisible by 32, so a(5) = a(4) + 16 = 19;
a(5)^5 - 3 = 2476096 which is divisible by 64, so a(6) = a(5) = 19.
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PROG
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(PARI) a(n) = lift(sqrtn(3+O(2^n), 5))
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CROSSREFS
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For the digits of 3^(1/5), see A325896.
Approximations of p-adic fifth-power roots:
this sequence (2-adic, 3^(1/5));
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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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