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A283754 The smallest number k such that k*2^n mod 3^n = 1.

%I A283754 #31 Mar 28 2017 14:44:51
%S A283754 2,7,17,76,38,262,1589,4075,11879,35464,17732,363160,181580,90790,
%T A283754 9611333,11980120,92083502,175181914,862431935,2174608168,1087304084,
%U A283754 543652042,271826021,235493860078,117746930039,1329806379184,664903189592,332451594796,166225797398,68713490263582,446139009321089
%N A283754 The smallest number k such that k*2^n mod 3^n = 1.
%C A283754 a(n) is the coefficient "a" in the Diophantine equation with two coefficients a and b, a * 2^n - b * 3^n = 1.
%H A283754 Robert Israel, <a href="/A283754/b283754.txt">Table of n, a(n) for n = 1..2095</a>
%F A283754 a(n) = ((3^n + 1)/2)^n mod 3^n (proved).
%F A283754 Conjecture: 2*a(n+1)-a(n) = 3^n * A055620(n). - _Robert Israel_, Mar 28 2017
%e A283754 2 * 2^1 mod 3^1 = 1, 7 * 2^2 mod 3^2 =1, 17 * 2^3 mod 3^3 = 1...
%p A283754 seq(2^(-n) mod 3^n, n=1..100); # _Robert Israel_, Mar 28 2017
%t A283754 Table[ PowerMod[ (3^n +1)/2, n, 3^n], {n, 30}] (* _Robert G. Wilson v_, Mar 28 2017 *)
%o A283754 (PARI) a(n)= my(z=3^n); lift( Mod((z + 1)/2, z)^n); \\ _Joerg Arndt_, Mar 24 2017
%Y A283754 Cf. A055620.
%K A283754 nonn
%O A283754 1,1
%A A283754 _Joe Slater_, Mar 23 2017
%E A283754 Corrected and more terms from _Joerg Arndt_, Mar 24 2017