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%I #48 May 03 2022 16:27:11
%S 3,3,5,6,9,53,9,36,12,33,9,186,21,33,111,144,9,564,3,330,239,273,3,
%T 1756,84,165,76,714,93,16167,21,5952,111,177,363,4288,21,15,99,5724,
%U 45,48807,45,4314,1140,183,9,14192,36,2940,495,1338,45,11572,747,11484
%N a(n) = |{m : multiplicative order of 10 mod m is equal to n}|.
%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).
%C 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
%C Also, number of primitive factors of 10^n - 1 (cf. A003060). - _Max Alekseyev_, May 03 2022
%C 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
%H <a href="/A059892/b059892.txt">Table of n, a(n) for n = 1..352</a>
%F 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).
%p with(numtheory):
%p a:= n-> add(mobius(n/d)*tau(10^d-1), d=divisors(n)):
%p seq(a(n), n=1..30); # _Alois P. Heinz_, Oct 12 2012
%t 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 *)
%o (PARI) j=[]; for(n=1,10,j=concat(j,sumdiv(n,d,moebius(n/d)*numdiv(10^d-1)))); j
%o (Python)
%o from sympy import divisors, mobius, divisor_count
%o def a(n): return sum(mobius(n//d)*divisor_count(10**d - 1) for d in divisors(n)) # _Indranil Ghosh_, Apr 23 2017
%Y 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).
%Y Cf. A000005, A008683, A002329, A007138, A005422, A057951, A058946, A070528.
%Y Column k=10 of A212957.
%K nonn,nice
%O 1,1
%A _Vladeta Jovovic_, Feb 06 2001
%E More terms from _Jason Earls_, Aug 06 2001.
%E Terms to a(280) in b-file from _T. D. Noe_, Oct 01 2013
%E a(281)-a(322) in b-file from _Ray Chandler_, May 03 2017
%E a(323)-a(352) in b-file from _Max Alekseyev_, May 03 2022
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