A322169 - OEIS

%I #36 Aug 28 2019 14:34:50

%S 2,4,1,3,1,4,1,4,1,0,3,2,3,3,1,1,3,1,1,3,3,3,3,1,4,1,2,4,2,0,0,1,0,0,

%T 3,0,3,2,1,3,0,0,3,2,4,1,1,0,3,3,2,2,3,0,2,0,3,3,3,1,2,0,2,0,0,0,0,3,

%U 3,0,0,2,0,3,1,1,0,4,1,0,4,0,4,0,3,4,0,3

%N Digits of the 5-adic integer 7^(1/5).

%C For k not divisible by 5, k is a fifth power in 5-adic field if and only if k == 1, 7, 18, 24 (mod 25). If k is a fifth power in 5-adic field, then k has exactly one fifth root.

%H Robert Israel, <a href="/A322169/b322169.txt">Table of n, a(n) for n = 0..10000</a>

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

%F a(n) = (A322157(n+1) - A322157(n))/5^n.

%e The unique number k in [1, 5^5] such that k^5 - 7 is divisible by 5^6 is k = 1047 = (13142)_5, so the first five terms are 2, 4, 1, 3 and 1.

%p op([1,3],padic:-rootp(x^5-7,5,100)); # _Robert Israel_, Aug 28 2019

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

%Y Cf. A322157.

%Y For fifth roots in 7-adic field, see A309445, A309446, A309447, A309448, A309449.

%K nonn,base

%O 0,1

%A _Jianing Song_, Aug 28 2019