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A327303 One of the two successive approximations up to 5^n for the 5-adic integer sqrt(-9). This is the 4 (mod 5) case (except for n = 0).

%I A327303 #19 Jan 17 2023 06:26:21
%S A327303 0,4,4,79,79,79,3204,18829,331329,1112579,1112579,20643829,118300079,
%T A327303 850721954,3292128204,27706190704,149776503204,302364393829,
%U A327303 1065303846954,8694698378204,46841671034454,332943965956329,332943965956329,5101315547987579,28943173458143829
%N A327303 One of the two successive approximations up to 5^n for the 5-adic integer sqrt(-9). This is the 4 (mod 5) case (except for n = 0).
%C A327303 a(n) is the unique number k in [1, 5^n] and congruent to 4 mod 5 such that k^2 + 9 is divisible by 5^n.
%H A327303 Robert Israel, <a href="/A327303/b327303.txt">Table of n, a(n) for n = 0..1424</a>
%H A327303 G. P. Michon, <a href="http://www.numericana.com/answer/p-adic.htm#integers">Introduction to p-adic integers</a>, Numericana.
%F A327303 a(1) = 4; for n >= 2, a(n) is the unique number k in {a(n-1) + m*5^(n-1) : m = 0, 1, 2, 3, 4} such that k^2 + 9 is divisible by 5^n.
%F A327303 For n > 0, a(n) = 5^n - A327302(n).
%e A327303 The unique number k in {4, 9, 14, 19, 24} such that k^2 + 9 is divisible by 25 is k = 4, so a(2) = 4.
%e A327303 The unique number k in {4, 29, 54, 79, 104} such that k^2 + 9 is divisible by 125 is k = 79, so a(3) = 46.
%e A327303 The unique number k in {79, 204, 329, 454, 579} such that k^2 + 9 is divisible by 625 is k = 79, so a(4) = 79.
%p A327303 R:= [padic:-rootp(x^2+9,5,101)]:
%p A327303 R:= op(select(t -> padic:-ratvaluep(t,1)=4, R)):
%p A327303 seq(padic:-ratvaluep(R,n),n=0..100); # _Robert Israel_, Jan 16 2023
%o A327303 (PARI) a(n) = truncate(-sqrt(-9+O(5^n)))
%Y A327303 For the digits of sqrt(-9) see A327304 and A327305.
%Y A327303 Approximations of 5-adic square roots:
%Y A327303 A327302, this sequence (sqrt(-9));
%Y A327303 A324027, A324028 (sqrt(-6));
%Y A327303 A268922, A269590 (sqrt(-4));
%Y A327303 A048898, A048899 (sqrt(-1));
%Y A327303 A324023, A324024 (sqrt(6)).
%K A327303 nonn
%O A327303 0,2
%A A327303 _Jianing Song_, Sep 16 2019