A337714 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.

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, 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, 3, 8, 1, 2, 4, 2, 4, 1

Offset

1,6

Comments

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.

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])

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

Links

Table of n, a(n) for n=1..76.

Formula

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.

Example

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

Mathematica

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

Prog

(PARI) a(n) = my(N=(3*n-1)\2); eulerphi(N)/znorder(Mod(3, N)); \\ Michel Marcus, Oct 22 2020

Crossrefs

Cf. A000010, A001651, A053446, A337936.

Sequence in context: A105246 A328027 A193829 * A198069 A300792 A132082

Adjacent sequences: A337711 A337712 A337713 * A337715 A337716 A337717

Keyword

nonn,easy

Author

Wolfdieter Lang, Oct 22 2020

Status

approved