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A059499 a(n) = |{m : multiplicative order of 2 mod m = n}|. 22
1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 3, 16, 1, 5, 5, 8, 1, 24, 1, 38, 9, 11, 3, 68, 6, 5, 4, 54, 7, 79, 1, 16, 11, 5, 13, 462, 3, 5, 13, 140, 3, 123, 7, 110, 54, 11, 7, 664, 2, 114, 29, 118, 7, 124, 59, 188, 13, 55, 3, 4456, 1, 5, 82, 96, 5, 353, 3, 118, 11, 485, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
Also, number of primitive factors of 2^n - 1 (cf. A212953). - Max Alekseyev, May 03 2022
The multiplicative order of a mod m, gcd(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m). See A002326.
a(n) is odd iff n is squarefree, A005117. - Thomas Ordowski, Jan 18 2014
The set S for which a(n) = |S| contains an odd number of prime powers p^k, where k > 0 and p == 3 (mod 4), iff n is squarefree and greater than one. - Isaac Saffold, Dec 28 2019
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..1206 (first 200 terms from Alois P. Heinz)
FORMULA
a(n) = Sum_{d|n} A008683(n/d) * A046801(d) = Sum_{d|A007947(n)} A008683(d) * A046801(n/d). - Max Alekseyev, May 03 2022
a(n) = 1 iff 2^n-1 is noncomposite. a(prime(n)) = 2^A088863(n)-1. - Thomas Ordowski, Jan 16 2014
EXAMPLE
a(3) = |{7}| = 1, a(4) = |{5,15}| = 2, a(6) = |{9,21,63}| = 3.
MAPLE
with(numtheory):
a:= n-> add(mobius(n/d)*tau(2^d-1), d=divisors(n)):
seq(a(n), n=1..100); # Alois P. Heinz, May 31 2012
MATHEMATICA
a[n_] := Sum[ MoebiusMu[n/d] * DivisorSigma[0, 2^d - 1], {d, Divisors[n]}]; Table[a[n], {n, 1, 71} ] (* Jean-François Alcover, Dec 12 2012 *)
CROSSREFS
Column k=2 of A212957.
Primitive factors of b^n - 1: this sequence (b=2), A059885 (b=3), A059886 (b=4), A059887 (b=5), A059888 (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).
Sequence in context: A100053 A029194 A246582 * A113322 A007380 A369813
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Feb 04 2001
EXTENSIONS
More terms from John W. Layman, Mar 22 2002
More terms from Alois P. Heinz, May 31 2012
STATUS
approved

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Last modified March 18 22:09 EDT 2024. Contains 370951 sequences. (Running on oeis4.)