%I #42 May 03 2022 10:42:08

%S 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,

%T 16,11,5,13,462,3,5,13,140,3,123,7,110,54,11,7,664,2,114,29,118,7,124,

%U 59,188,13,55,3,4456,1,5,82,96,5,353,3,118,11,485,7

%N a(n) = |{m : multiplicative order of 2 mod m = n}|.

%C Also, number of primitive factors of 2^n - 1 (cf. A212953). - _Max Alekseyev_, May 03 2022

%C 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.

%C a(n) is odd iff n is squarefree, A005117. - _Thomas Ordowski_, Jan 18 2014

%C 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

%H Max Alekseyev, <a href="/A059499/b059499.txt">Table of n, a(n) for n = 1..1206</a> (first 200 terms from Alois P. Heinz)

%F a(n) = Sum_{d|n} A008683(n/d) * A046801(d) = Sum_{d|A007947(n)} A008683(d) * A046801(n/d). - _Max Alekseyev_, May 03 2022

%F a(n) = 1 iff 2^n-1 is noncomposite. a(prime(n)) = 2^A088863(n)-1. - _Thomas Ordowski_, Jan 16 2014

%e a(3) = |{7}| = 1, a(4) = |{5,15}| = 2, a(6) = |{9,21,63}| = 3.

%p with(numtheory):

%p a:= n-> add(mobius(n/d)*tau(2^d-1), d=divisors(n)):

%p seq(a(n), n=1..100); # _Alois P. Heinz_, May 31 2012

%t 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 *)

%Y Column k=2 of A212957.

%Y 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).

%Y Cf. A001037, A046801, A058943, A059912, A112927, A212953.

%K nonn

%O 1,4

%A _Vladeta Jovovic_, Feb 04 2001

%E More terms from _John W. Layman_, Mar 22 2002

%E More terms from _Alois P. Heinz_, May 31 2012