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a(n) = Sum_{d|n} A007955(d) * A008683(n/d) = Sum_{d|n} A007955(d) * mu(n/d), where A007955(m) = number of divisors of m.
1

%I #13 Sep 08 2022 08:45:51

%S 1,1,2,6,4,32,6,56,24,94,10,1686,12,188,218,960,16,5772,18,7894,432,

%T 472,22,329992,120,662,702,21750,28,809648,30,31744,1076,1138,1214,

%U 10070172,36,1424,1506,2551944,40,3111034,42,84694,90876,2092,46,254471232,336,124780

%N a(n) = Sum_{d|n} A007955(d) * A008683(n/d) = Sum_{d|n} A007955(d) * mu(n/d), where A007955(m) = number of divisors of m.

%H Andrew Howroyd, <a href="/A174940/b174940.txt">Table of n, a(n) for n = 1..1000</a>

%F Moebius transform of A007955. - _Andrew Howroyd_, Jan 05 2020

%e For n = 4, A007955(n) = b(n): a(4) = b(1)*mu(4/1) + b(2)*mu(4/2) + b(4)*mu(4/4) = 1*0 + 2*(-1) + 8*1 = 6.

%t a[n_] := Sum[ MoebiusMu[n/d] * Times @@ Divisors[d], {d, Divisors[n]} ]; Table[ a[n], {n, 1, 30} ] (* _Jean-François Alcover_, Jan 09 2013 *)

%o (PARI) a(n)={sumdiv(n, d, vecprod(divisors(d))*moebius(n/d))} \\ _Andrew Howroyd_, Jan 05 2020

%o (Magma) [&+[&*Divisors(d)*MoebiusMu(n div d):d in Divisors(n)]:n in [1..50]]; // _Marius A. Burtea_, Jan 05 2020

%Y Cf. A008683 (mu), A007955 (product of divisors).

%K nonn

%O 1,3

%A _Jaroslav Krizek_, Apr 02 2010

%E Terms a(31) and beyond from _Andrew Howroyd_, Jan 05 2020