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A059892
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a(n) = |{m : multiplicative order of 10 mod m is equal to n}|.
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24
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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
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OFFSET
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1,1
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COMMENTS
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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
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
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LINKS
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FORMULA
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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).
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MAPLE
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with(numtheory):
a:= n-> add(mobius(n/d)*tau(10^d-1), d=divisors(n)):
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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Terms to a(280) in b-file from T. D. Noe, Oct 01 2013
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STATUS
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approved
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