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A322926
The successive approximations up to 2^n for 2-adic integer 5^(1/3).
5
0, 1, 1, 5, 13, 29, 29, 93, 93, 93, 605, 1629, 3677, 3677, 3677, 20061, 20061, 20061, 151133, 151133, 151133, 151133, 151133, 4345437, 4345437, 21122653, 54677085, 54677085, 188894813, 457330269, 457330269, 457330269, 2604813917, 6899781213, 6899781213
OFFSET
0,4
COMMENTS
a(n) is the unique solution to x^3 == 5 (mod 2^n) in the range [0, 2^n - 1].
FORMULA
For n > 0, a(n) = a(n-1) if a(n-1)^3 - 5 is divisible by 2^n, otherwise a(n-1) + 2^(n-1).
EXAMPLE
13^3 = 2197 = 137*2^4 + 5;
29^3 = 24389 = 762*2^5 + 5 = 381*2^6 + 5;
93^3 = 804357 = 6284*2^7 + 5 = 3142*2^8 + 5 = 1571*2^9 + 5.
PROG
(PARI) a(n) = lift(sqrtn(5+O(2^n), 3))
CROSSREFS
For the digits of 5^(1/3), see A323045.
Approximations of p-adic cubic roots:
A322701 (2-adic, 3^(1/3));
this sequence (2-adic, 5^(1/3));
A322934 (2-adic, 7^(1/3));
A322999 (2-adic, 9^(1/3));
A290567 (5-adic, 2^(1/3));
A290568 (5-adic, 3^(1/3));
A309444 (5-adic, 4^(1/3));
A319097, A319098, A319199 (7-adic, 6^(1/3));
A320914, A320915, A321105 (13-adic, 5^(1/3)).
Sequence in context: A226616 A226618 A321770 * A178854 A224339 A368546
KEYWORD
nonn
AUTHOR
Jianing Song, Aug 30 2019
STATUS
approved