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Moebius transform of repunits (A002275).
0

%I #7 Nov 05 2018 06:32:45

%S 1,10,110,1100,11110,110990,1111110,11110000,111111000,1111099990,

%T 11111111110,111110998900,1111111111110,11111109999990,

%U 111111111099890,1111111100000000,11111111111111110,111111110999889000,1111111111111111110,11111111109999998900,111111111111109999890

%N Moebius transform of repunits (A002275).

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>

%F G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = x/((1 - x)*(1 - 10*x)).

%F Dirichlet g.f.: (PolyLog(s,10) - zeta(s))/(9*zeta(s)), where PolyLog() is the polylogarithm function.

%F a(n) = Sum_{d|n} mu(n/d)*(10^d - 1)/9, where mu() is the Moebius function (A008683).

%t a[n_] := Sum[MoebiusMu[n/d] (10^d - 1)/9, {d, Divisors[n]}]; Array[a, 21]

%o (PARI) a(n) = sumdiv(n, d, moebius(n/d)*(10^d-1)/9); \\ _Michel Marcus_, Nov 05 2018

%Y Cf. A002275, A008683.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Apr 25 2017