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 A324077 One of the four successive approximations up to 13^n for 13-adic integer 3^(1/4).This is the 2 (mod 13) case (except for n = 0). 11
 0, 2, 28, 1211, 3408, 346140, 2573898, 60495606, 311489674, 6837335442, 70464331680, 208322823529, 18129926763899, 111322267253823, 1928572906807341, 29490207606702364, 438977351719428420, 7093143443551226830, 15743559362932564763, 1365208442786421282311 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS For n > 0, a(n) is the unique number k in [1, 13^n] and congruent to 2 mod 13 such that k^4 - 3 is divisible by 13^n. For k not divisible by 13, k is a fourth power in 13-adic field if and only if k == 1, 3, 9 (mod 13). If k is a fourth power in 13-adic field, then k has exactly 4 fourth-power roots. LINKS Wikipedia, p-adic number FORMULA a(n) = A324082(n)*A286840(n) mod 13^n = A324083(n)*A286841(n) mod 13^n. For n > 0, a(n) = 13^n - A324084(n). a(n)^2 == A322085(n) (mod 13^n). EXAMPLE The unique number k in [1, 13^2] and congruent to 2 modulo 13 such that k^4 - 3 is divisible by 13^2 is k = 28, so a(2) = 28. The unique number k in [1, 13^3] and congruent to 2 modulo 13 such that k^4 - 3 is divisible by 13^3 is k = 1211, so a(3) = 1211. PROG (PARI) a(n) = lift(sqrtn(3+O(13^n), 4) * sqrt(-1+O(13^n))) CROSSREFS Cf. A286840, A286841, A322085, A324082, A324083, A324084, A324085, A324086, A324087, A324153. Sequence in context: A300459 A009674 A143598 * A071220 A063794 A238817 Adjacent sequences:  A324074 A324075 A324076 * A324078 A324079 A324080 KEYWORD nonn AUTHOR Jianing Song, Sep 01 2019 STATUS approved

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Last modified January 23 07:07 EST 2020. Contains 331168 sequences. (Running on oeis4.)