A059892 a(n) = |{m : multiplicative order of 10 mod m is equal to n}|.

3, 3, 5, 6, 9, 53, 9, 36, 12, 33, 9, 186, 21, 33, 111, 144, 9, 564, 3, 330, 239, 273, 3, 1756, 84, 165, 76, 714, 93, 16167, 21, 5952, 111, 177, 363, 4288, 21, 15, 99, 5724, 45, 48807, 45, 4314, 1140, 183, 9, 14192, 36, 2940, 495, 1338, 45, 11572, 747, 11484

Offset: 1

Author

Vladeta Jovovic, Feb 06 2001

Keywords

• nonn
• nice

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).

The number of unit fractions 1/k having a decimal expansion of period n and with k coprime to 10. - T. D. Noe, May 18 2007

Also, number of primitive factors of 10^n - 1 (cf. A003060). - Max Alekseyev, May 03 2022

a(n) is odd if and only if n is squarefree. Proof: Note that 10^d - 1 == 3 (mod 4) for d >= 2, so 10^d - 1 is a square if and only if d = 1. From the formula we can see that a(n) is odd if and only if mu(n) is nonzero, or n is squarefree. - Jianing Song, Jun 15 2021

Links

• Table of n, a(n) for n = 1..352

Crossrefs

Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), A059886 (b=4), A059887 (b=5), A059888 (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), this sequence (b=10).

Cf. A000005, A008683, A002329, A007138, A005422, A057951, A058946, A070528.

Column k=10 of A212957.

Programs

Maple

with(numtheory):
a:= n-> add(mobius(n/d)*tau(10^d-1), d=divisors(n)):
seq(a(n), n=1..30);  # Alois P. Heinz, Oct 12 2012

Mathematica

f[n_, d_] := MoebiusMu[n/d]*Length[Divisors[10^d - 1]]; a[n_] := Total[(f[n, #] & ) /@ Divisors[n]]; Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Mar 21 2011 *)

PARI

j=[]; for(n=1,10,j=concat(j,sumdiv(n,d,moebius(n/d)*numdiv(10^d-1)))); j

Python

from sympy import divisors, mobius, divisor_count
def a(n): return sum(mobius(n//d)*divisor_count(10**d - 1) for d in divisors(n)) # Indranil Ghosh, Apr 23 2017

Formula

a(n) = Sum_{d|n} mu(n/d)*tau(10^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).

Extensions

More terms from Jason Earls, Aug 06 2001.

Terms to a(280) in b-file from T. D. Noe, Oct 01 2013

a(281)-a(322) in b-file from Ray Chandler, May 03 2017

a(323)-a(352) in b-file from Max Alekseyev, May 03 2022