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

2, 7, 17, 76, 38, 262, 1589, 4075, 11879, 35464, 17732, 363160, 181580, 90790, 9611333, 11980120, 92083502, 175181914, 862431935, 2174608168, 1087304084, 543652042, 271826021, 235493860078, 117746930039, 1329806379184, 664903189592, 332451594796, 166225797398, 68713490263582, 446139009321089

Offset: 1

Author

Joe Slater, Mar 23 2017

Keywords

• nonn

Comments

a(n) is the coefficient "a" in the Diophantine equation with two coefficients a and b, a * 2^n - b * 3^n = 1.

Examples

			2 * 2^1 mod 3^1 = 1, 7 * 2^2 mod 3^2 =1, 17 * 2^3 mod 3^3 = 1...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2095

Crossrefs

Cf. A055620.

Programs

Maple

seq(2^(-n) mod 3^n, n=1..100); # Robert Israel, Mar 28 2017

Mathematica

Table[ PowerMod[ (3^n +1)/2, n, 3^n], {n, 30}] (* Robert G. Wilson v, Mar 28 2017 *)

PARI

a(n)= my(z=3^n); lift( Mod((z + 1)/2, z)^n); \\ Joerg Arndt, Mar 24 2017

Formula

a(n) = ((3^n + 1)/2)^n mod 3^n (proved).

Conjecture: 2*a(n+1)-a(n) = 3^n * A055620(n). - Robert Israel, Mar 28 2017

Extensions

Corrected and more terms from Joerg Arndt, Mar 24 2017