A337714 - OEIS

%I #10 Nov 23 2020 02:06:53

%S 1,1,1,1,1,2,1,2,4,1,2,1,1,2,2,2,1,4,2,1,1,2,1,2,2,1,4,5,1,2,2,2,1,1,

%T 4,1,2,4,1,2,6,1,2,4,3,2,2,2,6,2,2,2,1,8,5,2,4,1,4,1,12,2,2,2,2,1,2,1,

%U 3,8,1,2,4,2,4,1

%N Euler totient function phi(N) divided by the multiplicative order of 3 modulo N, with N = N(n) = floor((3*n-1)/2), for n >= 1.

%C For the multiplicative order of 3 modulo N = N(n), with N(n) = floor((3*n-1)/2) = A001651(n), see A053446(n), for n >= 1.

%C For n >= 2 this sequence gives also the number of seeds s(N(n), i) needed to cover all numbers of the smallest positive restricted residue system (called RRS(N(n))) from the cycles obtained from s(N(n), i)*3^k (mod(N(n)), for k = 0..(P(N(n))-1), and certain s(N(n), i) chosen from RRS(N(n)). See A337936 for the choice of these seeds s(N, i). The cycles have period length P(N(n)) = A053446(n). For n = 1, N = 1, RRS(1) = [1] (not [0])

%C For the complete system of tripling sequences modulo N(n), for n >= 1, see A337936.

%F Bisection: a(2*k+1) = phi(3*k+1)/A053446(2*k+1), a(2*k+2) = phi(3*k+2)/A053446(2*k+2), for k >= 0, where phi = A000010.

%e The pairs [N(n),a(n)] begin, for n >= 1:

%e [1, 1], [2, 1], [4, 1], [5, 1], [7, 1], [8, 2], [10, 1], [11, 2], [13, 4], [14, 1], [16, 2], [17, 1], [19, 1], [20, 2], [22, 2], [23, 2], [25, 1], [26, 4], [28, 2], [29, 1], [31, 1], [32, 2], [34, 1], [35, 2], [37, 2], [38, 1], [40, 4], [41, 5], [43, 1], [44, 2], ...

%e The pairs [N(n)= floor((3*n-1)/2), P(N(n)) = A053446(n)] begin, for n >= 1:

%e [1, 1], [2, 1], [4, 2], [5, 4], [7, 6], [8, 2], [10, 4], [11, 5], [13, 3], [14, 6], [16, 4], [17, 16], [19, 18], [20, 4], [22, 5], [23, 11], [25, 20], [26, 3], [28, 6], [29, 28], [31, 30], [32, 8], [34, 16], [35, 12], [37, 18], [38, 18], [40, 4], [41, 8], [43, 42], [44, 10], ...

%t a[n_] := EulerPhi[(f = Floor[(3*n - 1)/2])] / MultiplicativeOrder[3, f]; Array[a, 100] (* _Amiram Eldar_, Oct 22 2020 *)

%o (PARI) a(n) = my(N=(3*n-1)\2); eulerphi(N)/znorder(Mod(3, N)); \\ _Michel Marcus_, Oct 22 2020

%Y Cf. A000010, A001651, A053446, A337936.

%K nonn,easy

%O 1,6

%A _Wolfdieter Lang_, Oct 22 2020