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A322701
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The successive approximations up to 2^n for 2-adic integer 3^(1/3).
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5
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0, 1, 3, 3, 11, 27, 59, 123, 123, 379, 379, 379, 379, 4475, 12667, 29051, 61819, 127355, 127355, 127355, 127355, 127355, 2224507, 2224507, 2224507, 19001723, 52556155, 119665019, 253882747, 253882747, 253882747, 1327624571, 3475108219, 7770075515
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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^3 == 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)^3 - 3 is divisible by 2^n, otherwise a(n-1) + 2^(n-1).
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EXAMPLE
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11^3 = 1331 = 83*2^4 + 3;
27^3 = 19683 = 615*2^5 + 3;
59^3 = 205379 = 3209*2^6 + 3.
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PROG
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(PARI) a(n) = lift(sqrtn(3+O(2^n), 3))
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CROSSREFS
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For the digits of 3^(1/3), see A323000.
Approximations of p-adic cubic roots:
this sequence (2-adic, 3^(1/3));
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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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