A325485 - OEIS

%I #7 Sep 11 2019 20:34:16

%S 0,2,22,22,397,397,6647,6647,319147,319147,6178522,6178522,103834772,

%T 592116022,3033522272,9137037897,70172194147,222760084772,

%U 3274517897272,3274517897272,60494976881647,441964703444147,1395639019850397,3779824810866022,51463540631178522

%N One of the four successive approximations up to 5^n for the 5-adic integer 6^(1/4). This is the 2 (mod 5) case (except for n = 0).

%C For n > 0, a(n) is the unique number k in [1, 5^n] and congruent to 2 mod 5 such that k^4 - 6 is divisible by 5^n.

%C For k not divisible by 5, k is a fourth power in 5-adic field if and only if k == 1 (mod 5). If k is a fourth power in 5-adic field, then k has exactly 4 fourth-power roots.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/P-adic_number">p-adic number</a>

%F a(n) = A325484(n)*A048898(n) mod 13^n = A325485(n)*A048899(n) mod 13^n.

%F For n > 0, a(n) = 5^n - A325486(n).

%F a(n)^2 == A324024(n) (mod 5^n).

%e The unique number k in [1, 5^2] and congruent to 2 modulo 5 such that k^4 - 6 is divisible by 5^2 is k = 22, so a(2) = 22.

%e The unique number k in [1, 5^3] and congruent to 2 modulo 5 such that k^4 - 6 is divisible by 5^3 is also k = 22, so a(3) is also 22.

%o (PARI) a(n) = lift(sqrtn(6+O(5^n), 4) * sqrt(-1+O(5^n)))

%Y Cf. A048898, A048899, A324024, A325489, A325490, A325491, A325492.

%Y Approximations of p-adic fourth-power roots:

%Y A325484, this sequence, A325486, A325487 (5-adic, 6^(1/4));

%Y A324077, A324082, A324083, A324084 (13-adic, 3^(1/4)).

%K nonn

%O 0,2

%A _Jianing Song_, Sep 07 2019