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A309451 The successive approximations up to 7^n for 7-adic integer 3^(1/5).

%I A309451 #10 Aug 03 2019 14:19:22
%S A309451 0,5,26,75,1104,3505,20312,20312,4961570,28020774,229788809,512264058,
%T A309451 2489590801,71696026806,71696026806,71696026806,19061942066578,
%U A309451 218459525484184,451090039471391
%N A309451 The successive approximations up to 7^n for 7-adic integer 3^(1/5).
%F A309451 a(0) = 0 and a(1) = 5, a(n) = a(n-1) + 2 * (a(n-1)^5 - 3) mod 7^n for n > 1.
%e A309451 a(1) = (   5)_7 = 5,
%e A309451 a(2) = (  35)_7 = 26,
%e A309451 a(3) = ( 135)_7 = 75,
%e A309451 a(4) = (3135)_7 = 1104.
%o A309451 (PARI) {a(n) = truncate((3+O(7^n))^(1/5))}
%Y A309451 Cf. A309446.
%Y A309451 Expansions of p-adic integers:
%Y A309451 A290568 (5-adic, 3^(1/3));
%Y A309451 A290800, A290802 (7-adic, sqrt(-6));
%Y A309451 A290806, A290809 (7-adic, sqrt(-5));
%Y A309451 A290803, A290804 (7-adic, sqrt(-3));
%Y A309451 A210852, A212153 (7-adic, (1+sqrt(-3))/2);
%Y A309451 A290557, A290559 (7-adic, sqrt(2));
%Y A309451 A309450 (7-adic, 2^(1/5));
%Y A309451 A309452 (7-adic, 4^(1/5));
%Y A309451 A309453 (7-adic, 5^(1/5));
%Y A309451 A309454 (7-adic, 6^(1/5)).
%K A309451 nonn
%O A309451 0,2
%A A309451 _Seiichi Manyama_, Aug 03 2019