A325894 - OEIS

%I #10 Sep 11 2019 20:37:25

%S 0,1,3,7,7,7,39,103,231,231,743,1767,1767,1767,1767,18151,18151,18151,

%T 18151,18151,542439,1591015,3688167,7882471,16271079,33048295,

%U 66602727,133711591,267929319,267929319,804800231,804800231,804800231,804800231,9394734823,26574604007

%N The successive approximations up to 2^n for the 2-adic integer 7^(1/5).

%C a(n) is the unique solution to x^5 == 7 (mod 2^n) in the range [0, 2^n - 1].

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

%F For n > 0, a(n) = a(n-1) if a(n-1)^5 - 7 is divisible by 2^n, otherwise a(n-1) + 2^(n-1).

%e For n = 2, the unique solution to x^5 == 7 (mod 4) in the range [0, 3] is x = 3, so a(2) = 3.

%e a(2)^5 - 7 = 236 which is not divisible by 8, so a(3) = a(2) + 4 = 7;

%e a(3)^5 - 7 = 16800 which is divisible by 16, so a(4) = a(3) = 7;

%e a(4)^5 - 7 = 16800 which is divisible by 32, so a(5) = a(4) = 7;

%e a(5)^5 - 7 = 16800 which is not divisible by 64, so a(6) = a(5) + 32 = 39.

%o (PARI) a(n) = lift(sqrtn(7+O(2^n), 5))

%Y For the digits of 7^(1/5), see A325898.

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

%Y A325892 (2-adic, 3^(1/5));

%Y A325893 (2-adic, 5^(1/5));

%Y this sequence (2-adic, 7^(1/5));

%Y A325895 (2-adic, 9^(1/5));

%Y A322157 (5-adic, 7^(1/5));

%Y A309450 (7-adic, 2^(1/5));

%Y A309451 (7-adic, 3^(1/5));

%Y A309452 (7-adic, 4^(1/5));

%Y A309453 (7-adic, 5^(1/5));

%Y A309454 (7-adic, 6^(1/5)).

%K nonn

%O 0,3

%A _Jianing Song_, Sep 07 2019