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A309454 The successive approximations up to 7^n for 7-adic integer 6^(1/5).

%I A309454 #11 Aug 03 2019 14:19:42
%S A309454 0,6,20,265,1980,11584,11584,246882,1070425,29894430,29894430,
%T A309454 1159795426,9069102398,9069102398,202847123212,2237516341759,
%U A309454 2237516341759,201635099759365,1132157155708193,6017397949439540,17416293134812683,496169890920484689,1613261619087052703
%N A309454 The successive approximations up to 7^n for 7-adic integer 6^(1/5).
%F A309454 a(0) = 0 and a(1) = 6, a(n) = a(n-1) + 4 * (a(n-1)^5 - 6) mod 7^n for n > 1.
%e A309454 a(1) = (   6)_7 = 6,
%e A309454 a(2) = (  26)_7 = 20,
%e A309454 a(3) = ( 526)_7 = 265,
%e A309454 a(4) = (5526)_7 = 1980.
%o A309454 (PARI) {a(n) = truncate((6+O(7^n))^(1/5))}
%Y A309454 Cf. A309449.
%Y A309454 Expansions of p-adic integers:
%Y A309454 A290800, A290802 (7-adic, sqrt(-6));
%Y A309454 A290806, A290809 (7-adic, sqrt(-5));
%Y A309454 A290803, A290804 (7-adic, sqrt(-3));
%Y A309454 A210852, A212153 (7-adic, (1+sqrt(-3))/2);
%Y A309454 A290557, A290559 (7-adic, sqrt(2));
%Y A309454 A309450 (7-adic, 2^(1/5));
%Y A309454 A309451 (7-adic, 3^(1/5));
%Y A309454 A309452 (7-adic, 4^(1/5));
%Y A309454 A309453 (7-adic, 5^(1/5)).
%K A309454 nonn
%O A309454 0,2
%A A309454 _Seiichi Manyama_, Aug 03 2019