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A059886 a(n) = |{m : multiplicative order of 4 mod m=n}|. 15
2, 2, 4, 4, 6, 16, 6, 8, 26, 38, 14, 68, 6, 54, 84, 16, 6, 462, 6, 140, 132, 110, 14, 664, 120, 118, 128, 188, 62, 4456, 6, 96, 364, 118, 498, 7608, 30, 118, 180, 568, 30, 9000, 30, 892, 3974, 494, 62, 5360, 24, 8024, 1524, 892, 62, 9600, 3050, 1784, 372, 446 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The multiplicative order of a mod m, GCD(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m).
a(n) is the number of orders of degree-n monic irreducible polynomials over GF(4).
Also, number of primitive factors of 4^n - 1. - Max Alekseyev, May 03 2022
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..1122 (first 160 terms from Alois P. Heinz)
FORMULA
a(n) = Sum_{ d divides n } mu(n/d)*tau(4^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).
EXAMPLE
a(1) = |{1,3}| = 2, a(2) = |{5,15}| =2, a(3) = |{7,9,21,63}| =4, a(4) = |{17,51,85,255}| = 4.
MAPLE
with(numtheory):
a:= n-> add(mobius(n/d)*tau(4^d-1), d=divisors(n)):
seq(a(n), n=1..60); # Alois P. Heinz, Oct 12 2012
MATHEMATICA
a[n_] := DivisorSum[n, MoebiusMu[n/#]*DivisorSigma[0, 4^# - 1]&]; Array[a, 100] (* Jean-François Alcover, Nov 11 2015 *)
CROSSREFS
Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), this sequence (b=4), A059887 (b=5), A059888 (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).
Column k=4 of A212957.
Sequence in context: A320194 A033732 A033752 * A272339 A267261 A219027
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Feb 06 2001
STATUS
approved

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Last modified March 18 22:34 EDT 2024. Contains 370951 sequences. (Running on oeis4.)